diff --git a/Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb b/Martinez_1D_Euler/.ipynb_checkpoints/1D_Euler_FINAL-May7-checkpoint.ipynb
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@@ -0,0 +1,1301 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:8ef564186e5a734ce19b023798a6e7e0115273b8c7d2b7bb859c83f62cf371a0"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "One-Dimensional Conservation Laws of Gas Dynamics "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "By Luis Martinez"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Latex\n",
+      "from IPython.display import Image"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The gas dynamics equations, which constitue the conservation of mass, momentum, and energy, can be used to study shock waves. This notebook includes a numerical scheme to solve for the classical Sod's Shock Tube problem.\n",
+      "\n",
+      "The Euler equations may be used to model other interesting physical phenomena. For example The Euler equations with a gravitational source term are used to model atmospheric pphenomena essential to weather predicion, climate modeling, and astrophisical modeling such as simulating solar climate and supernova explosions [Kappeli, Mishra, 2013]"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Principles of the Shock Tube Problem"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Initially, the tube is divided (by a diaphragm) into a high-pressure section (driver section) and a low pressure section (the driven section). There is no flow velocity anywhere inside the tube initially. \n",
+      "\n",
+      "As, the diaphragm is removed, the initial pressure discontinuity will propagate to the right as an unsteady normal shock wave traveling at velocity W. At the same time, an isentropic expansion wave will propagate to the left.\n",
+      "\n",
+      "We can see that shortly after the diaphragm is removed, there are 4 main regions in the tube.\n",
+      "\n",
+      "Region 1 is the undisturbed portion of the driven section (so we would expect initial conditions to still be true here)\n",
+      "\n",
+      "Region 2 :  This section has been \u2018processed\u2019 by the shock wave moving through it, and the pressure here is equal to the pressure behind the shock.\n",
+      "\n",
+      "Region 3: Has been \u2018processed\u2019 by the expansion wave moving through the region. The pressure here is equal to P3. Regions 2 and 3 are said to be at the same pressure since the gas in these regions cannot support any pressure discontinuities\n",
+      "\n",
+      "Region 4: The undisturbed portion of the driver section (we should expect initial conditions).\n",
+      "\n",
+      "It is interesting that Regions 2 and 3 are at the same velocity and pressure. But there are differences. Since region 2 has been processed by a shock wave and region 3 by an expansion wave, the entropy, temperature, and density are different. This is why we see this contact surface to differentiate the regions.\n",
+      "\n",
+      "Both Pressure and velocity change discontinuously across the shock wave. When we look at these properties across the expansion wave, we see that the variations of pressure and velocity are finite and continuous.\n",
+      "\n",
+      "This information was taken from Anderson(1995).\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='slide04.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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8EeF/hf8A8FUvAuk/DfUvE8PhvW/hD4kvL62ufGOo6rb3E1ve6WkUgF1cShWQ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\n+tfSZFmay6hWrr42uWP6s8HNMC8ZVp038KfM/lsj0Xw/8If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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 2,
+       "text": [
+        "<IPython.core.display.Image at 0x1022b0250>"
+       ]
+      }
+     ],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Governing Equations"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We look at one-dimensional conservation of mass, momentum, and energy:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\rho \\vec{v_x}}{\\partial x} = 0$$\n",
+      "\n",
+      "$$\\frac{\\partial \\rho \\vec{v_x}}{\\partial t} + \\frac{\\partial(\\rho \\vec{v_x}^2 +P)}{\\partial x} = 0$$\n",
+      "\n",
+      "$$\\frac{\\partial E}{\\partial t} +\\ \\frac{\\partial (E+P) \\vec{v_x}}{\\partial x} = 0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can also relate an equation of state:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$ P = (\\gamma-1)(E - \\frac{\\rho v^2}{2})$$"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "System of Equations"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "From the conservation Equations we can develop a system of equations:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial U}{\\partial t} + \\frac{\\partial f(U)}{\\partial x} = 0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "where U and f(U) are:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$ U = \\left(\n",
+      "\\begin{array}{c}\n",
+      "\\rho\\\\\n",
+      "\\rho \\vec{v_x}\\\\\n",
+      "E\n",
+      "\\end{array}\n",
+      "\\right)\n",
+      "\\,\\\n",
+      "f(U) = \\left(\n",
+      "\\begin{array}{c}\n",
+      "\\rho \\vec{v_x}\\\\\n",
+      "\\rho \\vec{v_x}^2 +\\ P\\\\\n",
+      "(E+P)\\vec{v_x}\n",
+      "\\end{array}\n",
+      "\\right)$$ "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Discretization of System of Equations: The Discontinuous Galerkin Method"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The discontinuous Galerkin method represents unknowns like the finite element method by piecewise polynomial functions, but unlike finite element methods, the polynomials are discontinuous at cell interfaces. At these interfaces, a flux is defined in the same way as for finite volume methods [Sonnendrucker, 2013. Numerical Methods for Hyperbolic Systems]\n",
+      "\n",
+      "On each cell, the unknown U is represented by a linear combination of basis functions of the space of polynomials of degree 'np'. The chosen basis function for this solver is a Legendre polynomial, and the interpolation points are Gauss-Lobatto points."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} W(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} W(x) dx = 0$$\n",
+      "\n",
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} \\phi_l(x) dx = 0$$\n",
+      "\n",
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + f(U)\\phi_l(x)\\ |_{I_i}\\ -\\ \\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx = 0$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Note that the Integral I goes from [-0.5] t0 [0.5]. From the Galerkin Method we can estimate U(x,t):"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$U_{i}(x,t) = \\sum_{0}^{np} U_{l}^{i}(t)\\phi_l(x)$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The objective here is to obtain an expression for U(t). Using orthogonality, we can define the spectral coefficients, which in turn allow us to define U(t):"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\alpha_l = \\int_{I_i} \\phi_l(x)\\ \\phi_l(x) dx\\\\$$\n",
+      "$$U_{l}^{i}(t) = \\frac{1}{\\alpha_l}\\ \\int_{I_i} U_{i}(x,t)\\ \\phi_l(x) dx\\\\$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Then, we can express the system of equations in a form that we can approximate:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial U_{l}^{i}(t)}{\\partial t} = \\frac{1}{\\alpha_l}  \n",
+      "\\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx \n",
+      "- f(U_{i+\\frac{1}{2}})\\phi_{i+\\frac{1}{2}}^l \n",
+      "+ f(U_{i-\\frac{1}{2}})\\phi_{i-\\frac{1}{2}}^l $$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The integral on the right-hand side can be approximated by Gauss-Lobatto quadratures and the fluxes are solved using the Rusanov flux. To march in time, a third-order runge-kutta method is used."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "The Solver"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The solver is tested by applying it to the classic Sod's shock tube problem. It is executed using a third order polynomial approximation, with a domain of 10 cells."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Import Libraries"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy\n",
+      "import sympy\n",
+      "from sympy.utilities.lambdify import lambdify\n",
+      "import math\n",
+      "from matplotlib import pyplot"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Import Scripts"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from Legendre_Polynomial_Function_np3 import *\n",
+      "from meshing import *\n",
+      "from U_initial_np3 import *\n",
+      "from Runge_Kutta_Third_Order_np3 import *\n",
+      "from Residual_Function_np3 import *\n",
+      "from Plot_Functions2 import *"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Some Program Parameters"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "np = 3 # order of scheme\n",
+      "ncell = 10 #number of cells\n",
+      "nface = ncell+1 #cell boundaries\n",
+      "rk3 = 3 #This is because we are solving 3 equations(mass, momentum, energy)\n",
+      "dt = 0.01\n",
+      "\n",
+      "# for flux function:\n",
+      "normal = 1.0"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Legendre Polynomial and Derivative"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fle_gauss = get_fle_gauss(np)\n",
+      "fle_grad_gauss = get_fle_derivative(np)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Meshing"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = get_x_cell(ncell)\n",
+      "dx = get_dx_cell(ncell)\n",
+      "f2c = get_face_to_cell(nface)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 7
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Initialize values of Mass, Momentum, and Energy"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='slide08.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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zSkEFgV9UYgfMW2gDknsO9fHuj/tgaNqt94R8Qv8ADPx3pvwW8VeJk8PeHPiP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\noAKKKKACiiigAooooAKKKKACiiigAooooA//2Q==\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 8,
+       "text": [
+        "<IPython.core.display.Image at 0x1022b0310>"
+       ]
+      }
+     ],
+     "prompt_number": 8
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The initial conditions are set as Sod's 1978 initial conditions. Using the mathematical relationships previously described, we can nitialize U, which contains mass, momentum, and energy:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "aic = get_inverse_mass_matrix(np)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "U_initial = get_U_initial(np, rk3, ncell, fle_gauss, x ,dx, aic)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 10
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Third Order Runge Kutta Time Scheme"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Stages of RK Scheme:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "\n",
+      "\n",
+      "$$U^1 = U^n\\ +\\ \\Delta R(U^n) \\\\$$\n",
+      "$$U^2 = \\frac{3}{4}U^n\\ +\\ \\frac{1}{4}\\ [U^1\\ +\\ \\Delta R(U^1)] \\\\$$\n",
+      "$$U^3 = \\frac{1}{3}U^n\\ +\\ \\frac{2}{3}\\ [U^2\\ +\\ \\Delta R(U^2)] \\\\$$\n"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Initialize Runge-Kutta Scheme:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Q_rk = U_initial"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Q_rk)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 12
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Boundary Conditions"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 13
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "First Cell Left Edge"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 14
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 15
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_left(Fl)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Last Cell Right Edge"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qfl = get_Qfl_right(Qf,np, ncell)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 17
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 18
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_right(rk3, ncell, Fl)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Interior Edge Flux"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 20
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "All Volume Flux"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "volflux = get_volflux(rk3, np, ncell, Qf)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Initialize h matrix"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Volume Inegral by Gauss Lobatto"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 23
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Compute The Residual Function"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 24
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Runge-Kutta Stages 1 through 3 for Initial Time"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qrk1 = get_Qrk_stage1(Q_rk, ncell, rk3, np, dt, residual)\n",
+      "\n",
+      "# Pre-Stage 2 RK, follow the steps as above to obtain new residual "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 25
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk1)\n",
+      "\n",
+      "#BC\n",
+      "\n",
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n",
+      "\n",
+      "#1st cell left edge\n",
+      "\n",
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n",
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n",
+      "bflux = get_bflux_left(Fl)\n",
+      "\n",
+      "#last cell right edge\n",
+      "\n",
+      "Qfl = get_Qfl_right(Qf,np, ncell)\n",
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n",
+      "bflux = get_bflux_right(rk3, ncell, Fl)\n",
+      "\n",
+      "#Interior Edge Flux\n",
+      "\n",
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n",
+      "volflux = get_volflux(rk3, np, ncell, Qf)\n",
+      "\n",
+      "#Initialize h matrix\n",
+      "\n",
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n",
+      "\n",
+      "# Volume Flux GL\n",
+      "\n",
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n",
+      "\n",
+      "#Compute residual (mass matrix)\n",
+      "\n",
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n",
+      "\n",
+      "#Stage 2 Rk--------------------------------------------------------------------\n",
+      "\n",
+      "Qrk2 = get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual)\n",
+      "\n",
+      "#Pre Stage 3 Rk-----------------------------------------------------------------\n",
+      "\n",
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk2)\n",
+      "\n",
+      "#BC\n",
+      "\n",
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n",
+      "\n",
+      "#1st cell left edge\n",
+      "\n",
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n",
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n",
+      "bflux = get_bflux_left(Fl)\n",
+      "\n",
+      "#last cell right edge\n",
+      "\n",
+      "Qfl = get_Qfl_right(Qf,np, ncell)\n",
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n",
+      "bflux = get_bflux_right(rk3, ncell, Fl)\n",
+      "\n",
+      "#Interior Edge Flux\n",
+      "\n",
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n",
+      "volflux = get_volflux(rk3, np, ncell, Qf)\n",
+      "\n",
+      "#Initialize h matrix\n",
+      "\n",
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n",
+      "\n",
+      "# Volume Flux GL\n",
+      "\n",
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n",
+      "\n",
+      "#Compute residual (mass matrix)\n",
+      "\n",
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n",
+      "\n",
+      "#Stage 3 Rk-------------------------------------------------------------\n",
+      "\n",
+      "Qrk3= get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 26
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Post-Processing"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Plots for Density, Velocity, Pressure, Energy"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x_density2, density2 = plot_density2(ncell,np, Qrk3, x, dx)\n",
+      "x_vel2, vel2 = plot_velocity2(ncell, np, Qrk3, x, dx)\n",
+      "x_pressure2, pressure2 = plot_pressure2(ncell,np, Qrk3, x, dx)\n",
+      "x_energy2, energy2 = plot_energy2(ncell,np, Qrk3, x, dx)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 27
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Density"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Density Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Density', fontsize=20)\n",
+      "pyplot.scatter(x_density2[20:30],density2[20:30])\n",
+      "pyplot.plot(x_density2[20:30],density2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"density.png\") ;"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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7Dmpr066i7BVyqqHZwHfMbCqwAagGrs9q05/cp40UkUrQcCCFtEohPd4rgFeA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\nAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1081a2350>"
+       ]
+      }
+     ],
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Velocity Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Velocity', fontsize=20)\n",
+      "pyplot.scatter(x_vel2[20:30],vel2[20:30]+1.)\n",
+      "pyplot.plot(x_vel2[20:30],vel2[20:30]+1, \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"velocity.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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aRFGyUmq8PyEk3duBrd19U3ff3d03BQYRupjtDTRXPkypukWL4Ec/Cn12V1st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\nYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x10830ce10>"
+       ]
+      }
+     ],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Pressure Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Pressure', fontsize=20)\n",
+      "pyplot.scatter(x_pressure2[20:30],pressure2[20:30])\n",
+      "pyplot.plot(x_pressure2[20:30],pressure2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"pressure.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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RabditFx3AFbF3UAbDgVedfdFUT/tvYSLHGb7LvAH4J0c66RUaSKBdGJxkuvL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\ngg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1083517d0>"
+       ]
+      }
+     ],
+     "prompt_number": 30
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "\n",
+      "pyplot.title('Energy Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Energy', fontsize=20)\n",
+      "pyplot.scatter(x_energy2[20:30],energy2[20:30])\n",
+      "pyplot.plot(x_energy2[20:30],energy2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"energy.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+8+bBOeekHYVIapRYJX66FbZUufYk1tY6i6ojqcAee8Brr4V7YIlUoVYHCJjZ\nRpony6ar/rk2NMDdvehzBWiAQInZYw+YM0eTsUjZS2qia8t6tLTcMpZLtVNzgFSxVucKSOOmgFIh\namvh00/TjkIkFZorQESkFUnf80pERPKgxCoiEjMlVhGRmCmxiojELPXEama/NLO3zOy5VspcY2aL\nzWyBmQ0tZnzSAfPnh4ECIlUm9cRKmL+1xTvPmdkYYJC71wATgWnFCkw66KabYNastKMQKbrUE6u7\nPwK820qRY4HpUdmngF5m1rsYsUkHNd0KW6TKpJ5Y89AHWJ7x+wqgb0qxSCE0+kqqVDkkVmg+TFYj\nAcqBaqxSpfK+/XWKVgL9Mn7vGy1rpq6ubtPPtbW11NbWJhmXtGXgQFi+PMxy1bVr2tGI5KWhoYGG\nhoYO7aMkhrSa2QDgD+7++RzrxgDnuPsYMxsGXOXuw3KU05DWUnT22XDppbDddmlHItIu7RnSmnpi\nNbPbgaOAHYG3gMlAVwB3vyEqcx2h58BHwCnu/nSO/SixikjsyjKxxkWJVUSSoElYRERKgBKriEjM\nlFhFRGKmxCrJmzZNNxaUqqLEKsm78kpYujTtKESKRolVkqehrVJllFgleRraKlVGiVWSpxqrVBkl\nVkmeaqxSZZRYJXlDh8KXv5x2FCJFoyGtIiKt0JBWEZESoMQqIhIzJVYRkZgpsYqIxEyJVYpjyRK4\n9tq0oxApCiVWKY5PPoGrr047CpGiUHcrKY5PPoFttoEPP9SNBaWsqLuVlK5u3aBPH81yJVVBiVWK\nR3MGSJVQYpXiUWKVKtEl7QCkipx8MnTRV04qny5eiYi0QhevRERKgBKriEjMlFhFRGKmxCoiEjMl\nVimu3/wGZsxIOwqRRCmxSnG9/TbMnZt2FCKJUmKV4qqp0Y0FpeIpsUpxafSVVAENEJDiaprl6oMP\nwsQsIiVOAwSk9HXrBn37apYrqWhKrFJ8t90Gu+ySdhQiiVFTgIhIK9QUICJSApRYRURipsQqIhKz\n1BOrmY0ys5fMbLGZXZhjfa2ZrTGzZ6LHRWnEKSKSr1QTq5l1Bq4DRgH7AF83s8E5is5z96HR49Ki\nBinJGD9eI7CkYqVdYz0EeNndl7r7euAO4Lgc5Qq6Iidl4L33YNGitKMQSUTaibUPsDzj9xXRskwO\nHG5mC8zsPjPbp2jRSXIGDdLQVqlYad/ZLZ+Op08D/dx9rZmNBmYBe+UqWFdXt+nn2tpaamtrYwhR\nEqE5A6QwAh4bAAAKCklEQVRENTQ00NDQ0KF9pDpAwMyGAXXuPir6/XvARnef0so2S4AD3X111nIN\nECgn990HV18N9fVpRyLSqnIcIPBXoMbMBphZN+BrwOzMAmbW28ws+vkQwj+D1c13JWVl0CBdvJKK\nlWpTgLtvMLNzgHqgM/ALd19oZmdE628ATgTOMrMNwFpgfGoBS3wGDoT77087CpFEaK4AEZFWlGNT\ngIhIxVFiFRGJmRKriEjMlFhFRGKmxCrpee01OOCAtKMQiZ16BUh61q+HrbfWjQWlpKlXgJSXrl2h\nXz9YsiTtSERipcQq6dJkLFKBlFglXTU1GtoqFUeJVdKlxFp09fX1jBgxjhEjxlGvSXASoYtXkq51\n60Jba+fOaUdSFerr6xk7dgKNjWECuR49LmTmzOmMHDky5chKly5eSfnp3j2xpKqaWXNTp94YJdUJ\nQEiwU6femHZYFSftia5FEpFdM3v00Qmx18zq6+s3JaVJkyaWxr4/+giWLYMVKz77GDMGjj++WfFv\n0cDub60MXd+6do0t/mqnxCoVqalm1puR7MuzfNL4Te750eWM7NsXttwSttsOevVq9/6TTNw59/37\nXzHy0EM3J8pdd4X992++8fXXwy9+AX37bn4ceCDsFW66MWnSRB59dAKNjaH4mm4zOdt3h913h9NO\ng9NPh/79O/weqp3aWCVVHar1ucM778Arr0CXLnDwwZtWjRgxjjlzjuVYenEu19CFleyy7Vvs3Xe3\nUKv76lfhyiub7/POO+Hyy2GrrUICbnoeORJOPrnZ/gdxOIfxBF2Yx+cHP8P53z4dPv0UhgyBo49u\nvv8nnoC77w5lNmzY/Dx8eLhzbca+YQL/zJ38mPPo3+kdum+9VUiU/frBSSeFRzvk/MwXLoQbboDb\nboPDD4eZM8NnKu1qY9UnJ6nZXDO7nO6sz6/WN38+XHwxvPpqeGyxBey5J5xwwmcSa1PNbHbjFGZz\ncrhI89s72LutxP2lL4WeCmvXhsdHH4XnPfbIWXwPlvAlHuRTltD7/XdhwYLQZrzzzrn3360b9OwZ\nklbnzpuf+2TfQzOYx1Ecz7nsfdQTzHpods4yhRo5cmTzz3jwYLjqKvjJT+CRR5RUO0g1VklNU83s\nbD6glgZ+S29GDZrHqbWHheTz05823+j11+Hxx0MyHTgwlGtB0m2gSV1dL9kr92vWwDbbQKfquubd\nnhqrEqukpimxHsQ+TOMslgEb+7/PV//zAviHf4Ajjkg7xFaV5MWrJF10Edx+O0ycCKec0nKtvMIo\nsVbIe6kWJVszk9zc4S9/gZ//HH7/exg9Gs48M7QPW0F5p6wosVbIe6kmJVkzk7a991640PWrX8Gc\nObD99kBl/j2VWCvkvYiUo0o9A1FirZD3IlKOMruJBdM55pjZPPDAjDTD6jANaRURKQHqrCYiscge\n1dWjx4VMmjQ93aBSoqYAEYmNLl5F21RKMlJiFZEkqI1VRKQEKLGKiMRMiVVEJGZKrCIiMVNiFRGJ\nmRKriEjMlFhFRGKmxCoiEjMlVhGRmCmxiojETIlVRCRmqSdWMxtlZi+Z2WIzu7CFMtdE6xeY2dBi\nxygiUohUE6uZdQauA0YB+wBfN7PBWWXGAIPcvQaYCEwreqAJamhoSDuEdivX2Ms1bijf2Ms17vZK\nu8Z6CPCyuy919/XAHcBxWWWOBaYDuPtTQC8z613cMJNTzl+4co29XOOG8o29XONur7QTax9gecbv\nK6JlbZXpm3BcIiLtlnZizXcC1ey5EDXxqoiUrFQnujazYUCdu4+Kfv8esNHdp2SU+TnQ4O53RL+/\nBBzl7m9l7UvJVkQSUehE12nf8+qvQI2ZDQBeB74GfD2rzGzgHOCOKBG/l51UofA3LiKSlFQTq7tv\nMLNzgHqgM/ALd19oZmdE629w9/vMbIyZvQx8BJySYsgiIm2qmHteiYiUirQvXhWsXAcUtBW3mf1L\nFO+zZvaYme2bRpzZ8vm8o3IHm9kGMzuhmPG1Js/vSq2ZPWNmz5tZQ5FDbFEe35cdzex+M5sfxf7N\nFMLMjumXZvaWmT3XSpmSOzah7dgLPj7dvWwehOaCl4EBQFdgPjA4q8wY4L7o50OBJ8sk7sOAntHP\no8ol7oxyDwH3AOPSjruAz7wX8ALQN/p9x7TjLiD2OuCypriBVUCXlOM+EhgKPNfC+pI7NguIvaDj\ns9xqrOU6oKDNuN39CXdfE/36FKXRVzefzxvg28BdwDvFDK4N+cT+DWCGu68AcPe/FznGluQT+xvA\nttHP2wKr3H1DEWNsxt0fAd5tpUgpHptA27EXenyWW2It1wEF+cSd6TTgvkQjyk+bcZtZH8JB3zTU\nuFQa7fP5zGuA7c1srpn91cz+tWjRtS6f2G8ChpjZ68AC4DtFiq0jSvHYbI82j8+0u1sVqlwHFOT9\n+mZ2NHAq8IXkwslbPnFfBfynu7uZGc0/+7TkE3tX4ADgH4EtgSfM7El3X5xoZG3LJ/bvA/PdvdbM\n9gTmmNl+7v5BwrF1VKkdmwXJ9/gst8S6EuiX8Xs/wn+91sr0jZalKZ+4iRrEbwJGuXtrp1TFkk/c\nBxL6GENo6xttZuvdfXZxQmxRPrEvB/7u7o1Ao5k9DOwHpJ1Y84n9cOC/ANz9FTNbAuxN6Bteqkrx\n2MxbQcdn2o3GBTYwdwFeITTqd6Pti1fDKIEG8jzj3p1wwWJY2vEWEndW+VuAE9KOu4DP/HPAg4SL\nRVsCzwH7lEnsPwMmRz/3JiTe7Usg9gHkd/GqJI7NAmIv6Pgsqxqrl+mAgnziBn4EbAdMi2p/6939\nkLRijuLKJ+6SlOd35SUzux94FtgI3OTuL6YXdZDn5/4T4BYzW0C4VvJdd1+dWtCAmd0OHAXsaGbL\ngcmE5paSPTabtBU7BR6fGiAgIhKzcusVICJS8pRYRURipsQqIhIzJVYRkZgpsYqIxEyJVUQkZkqs\nIiIxU2IVEYmZEquISMyUWEVEYqbEKhXPzGaZ2UYz+3aOdZdE625KIzapTJorQCqemW0HPEOYBeow\nd58fLf9H4AHgReBgd1+XXpRSSZRYpSqY2WHAPGAJYXLrrQnT8W1DSKoLUwxPKoyaAqQquPsTwA8J\nt2O5AbiNUIM9V0lV4qYaq1SVaP7VEdGvv3H3k9KMRyqTaqxSbWZGzw5cnWYgUrlUY5WqYWY1wNPA\nJ0BP4AXgEHf/ONXApOKoxipVwcy2AO4EegD/DFwGfJ5wl1mRWCmxSrX4KbA/cIW7/4lwT6PHgDPM\n7MRUI5OKo6YAqXhmNhaYATwJHOHuG6PlfQldrroAQ919SXpRSiVRYpWKZma7EwYHGLC/u7+Wtf5Y\nYBbwZ+BId19f/Cil0iixiojETG2sIiIxU2IVEYmZEquISMyUWEVEYqbEKiISMyVWEZGYKbGKiMRM\niVVEJGZKrCIiMVNiFRGJ2f8DUvytDsSbZJIAAAAASUVORK5CYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1084bdcd0>"
+       ]
+      }
+     ],
+     "prompt_number": 31
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Comparison with Gudunov's Method:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Sod (1978) examined several finite difference methods by appliying them to the shock tube problem. I have compared my results with the Gudunov method:"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Density"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide13.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 32,
+       "text": [
+        "<IPython.core.display.Image at 0x108138690>"
+       ]
+      }
+     ],
+     "prompt_number": 32
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Velocity"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide14.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 33,
+       "text": [
+        "<IPython.core.display.Image at 0x108138210>"
+       ]
+      }
+     ],
+     "prompt_number": 33
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Pressure"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide15.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 34,
+       "text": [
+        "<IPython.core.display.Image at 0x1081382d0>"
+       ]
+      }
+     ],
+     "prompt_number": 34
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Conclusions and On-going Work"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Several things can be done to further test the capabilities of the solver presented in this notebook. A more refined mesh would add a better understanding of localized behavior of the properties examined. \n",
+      "\n",
+      "Additionally, I would like to experiment with different time steps and see the effect on the stability of the system. Several numerical conditions should be improved, such as the handling of negative densities.\n",
+      "\n",
+      "As on-going work I will attempt to model reflective waves and move on to problems with source terms, eventually moving on to 2 and 3 dimensional problems"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "References"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "[Anderson, 1995]. Computational Fluid Dynamics, The Basics With Applications. John D. Anderson, Jr.1995.\n",
+      "\n",
+      "[Boyd,2001]. Chebyshev and Fourier Specral Methods. John P. Boyd. Second Edition, 2001.\n",
+      "\n",
+      "[Kappeli, Mishra, 2013]. Well Balanced Schemes for the Euler Equations with Gravitation. Journal of Computational Physics. R Kappeli, S. Mishra. 2013.\n",
+      "\n",
+      "[Liang]. Numerical Solution techniques in Mechanical and Aerospace Engineering. Discontinuous Galerkin Method for Hyperbolic PDEs. no date.\n",
+      "\n",
+      "[Sod, 1978].A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws. Gary A. Sod. 1978\n",
+      "\n",
+      "[Sonnendrucker, 2013]. Numerical Methods for Hyperbolic Systems. Max-Planck-Institut fur Plasmaphysik und Zentrum Mathematik, TU Munchen. July 2013."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.core.display import HTML\n",
+      "def css_styling():\n",
+      "    styles = open('../styles/custom.css', 'r').read()\n",
+      "    return HTML(styles)\n",
+      "css_styling()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "html": [
+        "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
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diff --git a/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb b/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb
new file mode 100644
index 0000000..a025b8d
--- /dev/null
+++ b/Martinez_1D_Euler/1D_Euler_FINAL-May7.ipynb
@@ -0,0 +1,1301 @@
+{
+ "metadata": {
+  "name": "",
+  "signature": "sha256:8ef564186e5a734ce19b023798a6e7e0115273b8c7d2b7bb859c83f62cf371a0"
+ },
+ "nbformat": 3,
+ "nbformat_minor": 0,
+ "worksheets": [
+  {
+   "cells": [
+    {
+     "cell_type": "heading",
+     "level": 1,
+     "metadata": {},
+     "source": [
+      "One-Dimensional Conservation Laws of Gas Dynamics "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "By Luis Martinez"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.display import Latex\n",
+      "from IPython.display import Image"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 1
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The gas dynamics equations, which constitue the conservation of mass, momentum, and energy, can be used to study shock waves. This notebook includes a numerical scheme to solve for the classical Sod's Shock Tube problem.\n",
+      "\n",
+      "The Euler equations may be used to model other interesting physical phenomena. For example The Euler equations with a gravitational source term are used to model atmospheric pphenomena essential to weather predicion, climate modeling, and astrophisical modeling such as simulating solar climate and supernova explosions [Kappeli, Mishra, 2013]"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Principles of the Shock Tube Problem"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Initially, the tube is divided (by a diaphragm) into a high-pressure section (driver section) and a low pressure section (the driven section). There is no flow velocity anywhere inside the tube initially. \n",
+      "\n",
+      "As, the diaphragm is removed, the initial pressure discontinuity will propagate to the right as an unsteady normal shock wave traveling at velocity W. At the same time, an isentropic expansion wave will propagate to the left.\n",
+      "\n",
+      "We can see that shortly after the diaphragm is removed, there are 4 main regions in the tube.\n",
+      "\n",
+      "Region 1 is the undisturbed portion of the driven section (so we would expect initial conditions to still be true here)\n",
+      "\n",
+      "Region 2 :  This section has been \u2018processed\u2019 by the shock wave moving through it, and the pressure here is equal to the pressure behind the shock.\n",
+      "\n",
+      "Region 3: Has been \u2018processed\u2019 by the expansion wave moving through the region. The pressure here is equal to P3. Regions 2 and 3 are said to be at the same pressure since the gas in these regions cannot support any pressure discontinuities\n",
+      "\n",
+      "Region 4: The undisturbed portion of the driver section (we should expect initial conditions).\n",
+      "\n",
+      "It is interesting that Regions 2 and 3 are at the same velocity and pressure. But there are differences. Since region 2 has been processed by a shock wave and region 3 by an expansion wave, the entropy, temperature, and density are different. This is why we see this contact surface to differentiate the regions.\n",
+      "\n",
+      "Both Pressure and velocity change discontinuously across the shock wave. When we look at these properties across the expansion wave, we see that the variations of pressure and velocity are finite and continuous.\n",
+      "\n",
+      "This information was taken from Anderson(1995).\n"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='slide04.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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8EeF/hf8A8FUvAuk/DfUvE8PhvW/hD4kvL62ufGOo6rb3E1ve6WkUgF1cShWQ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\n+tfSZFmay6hWrr42uWP6s8HNMC8ZVp038KfM/lsj0Xw/8If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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 2,
+       "text": [
+        "<IPython.core.display.Image at 0x1022b0250>"
+       ]
+      }
+     ],
+     "prompt_number": 2
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Governing Equations"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We look at one-dimensional conservation of mass, momentum, and energy:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial \\rho}{\\partial t} + \\frac{\\partial \\rho \\vec{v_x}}{\\partial x} = 0$$\n",
+      "\n",
+      "$$\\frac{\\partial \\rho \\vec{v_x}}{\\partial t} + \\frac{\\partial(\\rho \\vec{v_x}^2 +P)}{\\partial x} = 0$$\n",
+      "\n",
+      "$$\\frac{\\partial E}{\\partial t} +\\ \\frac{\\partial (E+P) \\vec{v_x}}{\\partial x} = 0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "We can also relate an equation of state:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$ P = (\\gamma-1)(E - \\frac{\\rho v^2}{2})$$"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "System of Equations"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "From the conservation Equations we can develop a system of equations:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial U}{\\partial t} + \\frac{\\partial f(U)}{\\partial x} = 0$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "where U and f(U) are:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$ U = \\left(\n",
+      "\\begin{array}{c}\n",
+      "\\rho\\\\\n",
+      "\\rho \\vec{v_x}\\\\\n",
+      "E\n",
+      "\\end{array}\n",
+      "\\right)\n",
+      "\\,\\\n",
+      "f(U) = \\left(\n",
+      "\\begin{array}{c}\n",
+      "\\rho \\vec{v_x}\\\\\n",
+      "\\rho \\vec{v_x}^2 +\\ P\\\\\n",
+      "(E+P)\\vec{v_x}\n",
+      "\\end{array}\n",
+      "\\right)$$ "
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Discretization of System of Equations: The Discontinuous Galerkin Method"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The discontinuous Galerkin method represents unknowns like the finite element method by piecewise polynomial functions, but unlike finite element methods, the polynomials are discontinuous at cell interfaces. At these interfaces, a flux is defined in the same way as for finite volume methods [Sonnendrucker, 2013. Numerical Methods for Hyperbolic Systems]\n",
+      "\n",
+      "On each cell, the unknown U is represented by a linear combination of basis functions of the space of polynomials of degree 'np'. The chosen basis function for this solver is a Legendre polynomial, and the interpolation points are Gauss-Lobatto points."
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} W(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} W(x) dx = 0$$\n",
+      "\n",
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + \\int_{I_i} \\frac{\\partial f(U(x,t))}{\\partial x} \\phi_l(x) dx = 0$$\n",
+      "\n",
+      "$$\\int_{I_i} \\frac{\\partial U(x,t)}{\\partial t} \\phi_l(x)\\ dx\\ + f(U)\\phi_l(x)\\ |_{I_i}\\ -\\ \\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx = 0$$\n"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Note that the Integral I goes from [-0.5] t0 [0.5]. From the Galerkin Method we can estimate U(x,t):"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$U_{i}(x,t) = \\sum_{0}^{np} U_{l}^{i}(t)\\phi_l(x)$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The objective here is to obtain an expression for U(t). Using orthogonality, we can define the spectral coefficients, which in turn allow us to define U(t):"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\alpha_l = \\int_{I_i} \\phi_l(x)\\ \\phi_l(x) dx\\\\$$\n",
+      "$$U_{l}^{i}(t) = \\frac{1}{\\alpha_l}\\ \\int_{I_i} U_{i}(x,t)\\ \\phi_l(x) dx\\\\$$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Then, we can express the system of equations in a form that we can approximate:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "$$\\frac{\\partial U_{l}^{i}(t)}{\\partial t} = \\frac{1}{\\alpha_l}  \n",
+      "\\int_{I_i} f(U)\\frac{\\partial \\phi_l(x)}{\\partial x} dx \n",
+      "- f(U_{i+\\frac{1}{2}})\\phi_{i+\\frac{1}{2}}^l \n",
+      "+ f(U_{i-\\frac{1}{2}})\\phi_{i-\\frac{1}{2}}^l $$"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The integral on the right-hand side can be approximated by Gauss-Lobatto quadratures and the fluxes are solved using the Rusanov flux. To march in time, a third-order runge-kutta method is used."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "The Solver"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The solver is tested by applying it to the classic Sod's shock tube problem. It is executed using a third order polynomial approximation, with a domain of 10 cells."
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Import Libraries"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "import numpy\n",
+      "import sympy\n",
+      "from sympy.utilities.lambdify import lambdify\n",
+      "import math\n",
+      "from matplotlib import pyplot"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 3
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Import Scripts"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from Legendre_Polynomial_Function_np3 import *\n",
+      "from meshing import *\n",
+      "from U_initial_np3 import *\n",
+      "from Runge_Kutta_Third_Order_np3 import *\n",
+      "from Residual_Function_np3 import *\n",
+      "from Plot_Functions2 import *"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 4
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Some Program Parameters"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "np = 3 # order of scheme\n",
+      "ncell = 10 #number of cells\n",
+      "nface = ncell+1 #cell boundaries\n",
+      "rk3 = 3 #This is because we are solving 3 equations(mass, momentum, energy)\n",
+      "dt = 0.01\n",
+      "\n",
+      "# for flux function:\n",
+      "normal = 1.0"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 5
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Legendre Polynomial and Derivative"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "fle_gauss = get_fle_gauss(np)\n",
+      "fle_grad_gauss = get_fle_derivative(np)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 6
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Meshing"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x = get_x_cell(ncell)\n",
+      "dx = get_dx_cell(ncell)\n",
+      "f2c = get_face_to_cell(nface)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 7
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Initialize values of Mass, Momentum, and Energy"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='slide08.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
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zSkEFgV9UYgfMW2gDknsO9fHuj/tgaNqt94R8Qv8ADPx3pvwW8VeJk8PeHPiP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\noAKKKKACiiigAooooAKKKKACiiigAooooA//2Q==\n",
+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 8,
+       "text": [
+        "<IPython.core.display.Image at 0x1022b0310>"
+       ]
+      }
+     ],
+     "prompt_number": 8
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "The initial conditions are set as Sod's 1978 initial conditions. Using the mathematical relationships previously described, we can nitialize U, which contains mass, momentum, and energy:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "aic = get_inverse_mass_matrix(np)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 9
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "U_initial = get_U_initial(np, rk3, ncell, fle_gauss, x ,dx, aic)\n"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 10
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Third Order Runge Kutta Time Scheme"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Stages of RK Scheme:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "\n",
+      "\n",
+      "$$U^1 = U^n\\ +\\ \\Delta R(U^n) \\\\$$\n",
+      "$$U^2 = \\frac{3}{4}U^n\\ +\\ \\frac{1}{4}\\ [U^1\\ +\\ \\Delta R(U^1)] \\\\$$\n",
+      "$$U^3 = \\frac{1}{3}U^n\\ +\\ \\frac{2}{3}\\ [U^2\\ +\\ \\Delta R(U^2)] \\\\$$\n"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Initialize Runge-Kutta Scheme:"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Q_rk = U_initial"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 11
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Q_rk)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 12
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Boundary Conditions"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 13
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "First Cell Left Edge"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 14
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 15
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_left(Fl)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 16
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Last Cell Right Edge"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qfl = get_Qfl_right(Qf,np, ncell)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 17
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 18
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_right(rk3, ncell, Fl)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 19
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Interior Edge Flux"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 20
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "All Volume Flux"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "volflux = get_volflux(rk3, np, ncell, Qf)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 21
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Initialize h matrix"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 22
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Volume Inegral by Gauss Lobatto"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 23
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Compute The Residual Function"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 24
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Runge-Kutta Stages 1 through 3 for Initial Time"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qrk1 = get_Qrk_stage1(Q_rk, ncell, rk3, np, dt, residual)\n",
+      "\n",
+      "# Pre-Stage 2 RK, follow the steps as above to obtain new residual "
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 25
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk1)\n",
+      "\n",
+      "#BC\n",
+      "\n",
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n",
+      "\n",
+      "#1st cell left edge\n",
+      "\n",
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n",
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n",
+      "bflux = get_bflux_left(Fl)\n",
+      "\n",
+      "#last cell right edge\n",
+      "\n",
+      "Qfl = get_Qfl_right(Qf,np, ncell)\n",
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n",
+      "bflux = get_bflux_right(rk3, ncell, Fl)\n",
+      "\n",
+      "#Interior Edge Flux\n",
+      "\n",
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n",
+      "volflux = get_volflux(rk3, np, ncell, Qf)\n",
+      "\n",
+      "#Initialize h matrix\n",
+      "\n",
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n",
+      "\n",
+      "# Volume Flux GL\n",
+      "\n",
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n",
+      "\n",
+      "#Compute residual (mass matrix)\n",
+      "\n",
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n",
+      "\n",
+      "#Stage 2 Rk--------------------------------------------------------------------\n",
+      "\n",
+      "Qrk2 = get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual)\n",
+      "\n",
+      "#Pre Stage 3 Rk-----------------------------------------------------------------\n",
+      "\n",
+      "Qf = get_Qf(ncell, rk3, np, fle_gauss, Qrk2)\n",
+      "\n",
+      "#BC\n",
+      "\n",
+      "Qbleft, Qbright = get_Qbright_Qbleft(rk3,np, ncell, Qf)\n",
+      "\n",
+      "#1st cell left edge\n",
+      "\n",
+      "Qfl, Qfr = get_Qfl_Qfr_left(rk3, Qf, np, Qbleft)\n",
+      "Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)\n",
+      "bflux = get_bflux_left(Fl)\n",
+      "\n",
+      "#last cell right edge\n",
+      "\n",
+      "Qfl = get_Qfl_right(Qf,np, ncell)\n",
+      "Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)\n",
+      "bflux = get_bflux_right(rk3, ncell, Fl)\n",
+      "\n",
+      "#Interior Edge Flux\n",
+      "\n",
+      "bflux = get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux)\n",
+      "volflux = get_volflux(rk3, np, ncell, Qf)\n",
+      "\n",
+      "#Initialize h matrix\n",
+      "\n",
+      "h = get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss)\n",
+      "\n",
+      "# Volume Flux GL\n",
+      "\n",
+      "h = get_GL_hmatrix(rk3, np, ncell, h, volflux,volcoef, fle_grad_gauss)\n",
+      "\n",
+      "#Compute residual (mass matrix)\n",
+      "\n",
+      "residual = get_residual_matrix(rk3, np, ncell, aic, h, dx)\n",
+      "\n",
+      "#Stage 3 Rk-------------------------------------------------------------\n",
+      "\n",
+      "Qrk3= get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 26
+    },
+    {
+     "cell_type": "heading",
+     "level": 3,
+     "metadata": {},
+     "source": [
+      "Post-Processing"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Plots for Density, Velocity, Pressure, Energy"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "x_density2, density2 = plot_density2(ncell,np, Qrk3, x, dx)\n",
+      "x_vel2, vel2 = plot_velocity2(ncell, np, Qrk3, x, dx)\n",
+      "x_pressure2, pressure2 = plot_pressure2(ncell,np, Qrk3, x, dx)\n",
+      "x_energy2, energy2 = plot_energy2(ncell,np, Qrk3, x, dx)"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [],
+     "prompt_number": 27
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Density"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Density Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Density', fontsize=20)\n",
+      "pyplot.scatter(x_density2[20:30],density2[20:30])\n",
+      "pyplot.plot(x_density2[20:30],density2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"density.png\") ;"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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7Dmpr066i7BVyqqHZwHfMbCqwAagGrs9q05/cp40UkUrQcCCFtEohPd4rgFeA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\nAABJRU5ErkJggg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1081a2350>"
+       ]
+      }
+     ],
+     "prompt_number": 28
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Velocity Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Velocity', fontsize=20)\n",
+      "pyplot.scatter(x_vel2[20:30],vel2[20:30]+1.)\n",
+      "pyplot.plot(x_vel2[20:30],vel2[20:30]+1, \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"velocity.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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aRFGyUmq8PyEk3duBrd19U3ff3d03BQYRupjtDTRXPkypukWL4Ec/Cn12V1st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\nYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x10830ce10>"
+       ]
+      }
+     ],
+     "prompt_number": 29
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "pyplot.title('Pressure Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Pressure', fontsize=20)\n",
+      "pyplot.scatter(x_pressure2[20:30],pressure2[20:30])\n",
+      "pyplot.plot(x_pressure2[20:30],pressure2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"pressure.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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RabditFx3AFbF3UAbDgVedfdFUT/tvYSLHGb7LvAH4J0c66RUaSKBdGJxkuvL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\ngg==\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1083517d0>"
+       ]
+      }
+     ],
+     "prompt_number": 30
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "%matplotlib inline\n",
+      "\n",
+      "pyplot.figure(figsize=(5,5))\n",
+      "\n",
+      "pyplot.title('Energy Profile at t = 0.01 s', fontsize=20)\n",
+      "pyplot.xlabel('x', fontsize=20)\n",
+      "pyplot.ylabel('Energy', fontsize=20)\n",
+      "pyplot.scatter(x_energy2[20:30],energy2[20:30])\n",
+      "pyplot.plot(x_energy2[20:30],energy2[20:30], \n",
+      "            linewidth=1, color = 'r', linestyle='--')\n",
+      "\n",
+      "pyplot.savefig(\"energy.png\");"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "metadata": {},
+       "output_type": "display_data",
+       "png": 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+8+bBOeekHYVIapRYJX66FbZUufYk1tY6i6ojqcAee8Brr4V7YIlUoVYHCJjZ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\niVVEJGZKrCIiMVNiFRGJ2f8DUvytDsSbZJIAAAAASUVORK5CYII=\n",
+       "text": [
+        "<matplotlib.figure.Figure at 0x1084bdcd0>"
+       ]
+      }
+     ],
+     "prompt_number": 31
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Comparison with Gudunov's Method:"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Sod (1978) examined several finite difference methods by appliying them to the shock tube problem. I have compared my results with the Gudunov method:"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Density"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide13.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 32,
+       "text": [
+        "<IPython.core.display.Image at 0x108138690>"
+       ]
+      }
+     ],
+     "prompt_number": 32
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Velocity"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide14.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 33,
+       "text": [
+        "<IPython.core.display.Image at 0x108138210>"
+       ]
+      }
+     ],
+     "prompt_number": 33
+    },
+    {
+     "cell_type": "heading",
+     "level": 5,
+     "metadata": {},
+     "source": [
+      "Pressure"
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "Image(filename='Slide15.jpg')"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "jpeg": 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+       "metadata": {},
+       "output_type": "pyout",
+       "prompt_number": 34,
+       "text": [
+        "<IPython.core.display.Image at 0x1081382d0>"
+       ]
+      }
+     ],
+     "prompt_number": 34
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "Conclusions and On-going Work"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "Several things can be done to further test the capabilities of the solver presented in this notebook. A more refined mesh would add a better understanding of localized behavior of the properties examined. \n",
+      "\n",
+      "Additionally, I would like to experiment with different time steps and see the effect on the stability of the system. Several numerical conditions should be improved, such as the handling of negative densities.\n",
+      "\n",
+      "As on-going work I will attempt to model reflective waves and move on to problems with source terms, eventually moving on to 2 and 3 dimensional problems"
+     ]
+    },
+    {
+     "cell_type": "heading",
+     "level": 2,
+     "metadata": {},
+     "source": [
+      "References"
+     ]
+    },
+    {
+     "cell_type": "markdown",
+     "metadata": {},
+     "source": [
+      "[Anderson, 1995]. Computational Fluid Dynamics, The Basics With Applications. John D. Anderson, Jr.1995.\n",
+      "\n",
+      "[Boyd,2001]. Chebyshev and Fourier Specral Methods. John P. Boyd. Second Edition, 2001.\n",
+      "\n",
+      "[Kappeli, Mishra, 2013]. Well Balanced Schemes for the Euler Equations with Gravitation. Journal of Computational Physics. R Kappeli, S. Mishra. 2013.\n",
+      "\n",
+      "[Liang]. Numerical Solution techniques in Mechanical and Aerospace Engineering. Discontinuous Galerkin Method for Hyperbolic PDEs. no date.\n",
+      "\n",
+      "[Sod, 1978].A Survey of Several Finite Difference Methods for Systems of Nonlinear Hyperbolic Conservation Laws. Gary A. Sod. 1978\n",
+      "\n",
+      "[Sonnendrucker, 2013]. Numerical Methods for Hyperbolic Systems. Max-Planck-Institut fur Plasmaphysik und Zentrum Mathematik, TU Munchen. July 2013."
+     ]
+    },
+    {
+     "cell_type": "code",
+     "collapsed": false,
+     "input": [
+      "from IPython.core.display import HTML\n",
+      "def css_styling():\n",
+      "    styles = open('../styles/custom.css', 'r').read()\n",
+      "    return HTML(styles)\n",
+      "css_styling()"
+     ],
+     "language": "python",
+     "metadata": {},
+     "outputs": [
+      {
+       "html": [
+        "<link href='http://fonts.googleapis.com/css?family=Fenix' rel='stylesheet' type='text/css'>\n",
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diff --git a/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py
new file mode 100644
index 0000000..fc333e3
--- /dev/null
+++ b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.py
@@ -0,0 +1,166 @@
+# Legendre Polynomial Function
+
+import sympy
+import numpy
+from sympy.utilities.lambdify import lambdify
+
+#-------------------------------------------------------
+np = 3
+xp = sympy.symbols('xp')
+#-------------------------------------------------------
+
+# Store the Quadrature Points and Weights for 4 point quadrature(np=3), fifth order polynomial:
+
+#Quadrature Points
+
+gauss = numpy.zeros((np+1, np+1), dtype=float)
+gauss[0,0]=-0.5
+gauss[1,0]=0.5
+gauss[2,0]= -1 * numpy.sqrt(5)/10
+gauss[3,0]= numpy.sqrt(5)/10.0
+
+# Corresponding Quadrature weights
+
+gauss[0,1]=1.0/12.0
+gauss[1,1]=gauss[0,1]
+gauss[2,1]=5.0/12.0
+gauss[3,1]=gauss[2,1]
+
+#----------------------------------------------------------
+#Legendre Polynomial Terms
+fle = [1.0]*(np+1)
+
+fle[0] = 1.0
+fle[1] = xp
+fle[2] = (xp**2)-(1.0/12.0)
+fle[3] = (xp**3)-(0.15*xp)
+ 
+#---------------------------------------------------------
+fle_eval = [1.0]*(np+1)
+
+for i in range(np+1):
+	fle_eval[i] = lambdify((xp),fle[i])
+#----------------------------------------------------------
+fle_0_eval = fle_eval[0]
+fle_1_eval = fle_eval[1]
+fle_2_eval = fle_eval[2]
+fle_3_eval = fle_eval[3]
+
+#----------------------------------------------------------
+def fle0(xp):
+	f0 = fle_0_eval(xp)
+	return f0
+
+def fle1(xp):
+	f1 = fle_1_eval(xp)
+	return f1
+
+def fle2(xp):
+	f2 = fle_2_eval(xp)
+	return f2
+    
+def fle3(xp):
+	f3 = fle_3_eval(xp)
+	return f3
+
+#----------------------------------------------------------------------	
+#Legendre Polynomial Derivative
+	
+fle_grad = [0.0]*(np+1)
+fle_grad[0] = 0.0
+fle_grad[1] = 1
+fle_grad[2] = 2*xp
+fle_grad[3] = (3*(xp**2))-(0.15) 
+
+#--------------------------------------------------------------------
+fle_grad_eval = [1.0]*(np+1)
+
+for i in range(np+1):
+	fle_grad_eval[i] = lambdify((xp),fle_grad[i])
+#--------------------------------------------------------------------
+fle_grad_0_eval = fle_grad_eval[0]
+fle_grad_1_eval = fle_grad_eval[1]
+fle_grad_2_eval = fle_grad_eval[2]
+fle_grad_3_eval = fle_grad_eval[3]
+
+
+#--------------------------------------------------------------------
+def fle0_grad(xp):
+	fg0 = fle_grad_0_eval(xp)
+	return fg0
+
+def fle1_grad(xp):
+	fg1 = fle_grad_1_eval(xp)
+	return fg1
+
+def fle2_grad(xp):
+	fg2 = fle_grad_2_eval(xp)
+	return fg2
+    
+def fle3_grad(xp):
+	fg3 = fle_grad_3_eval(xp)
+	return fg3
+
+#----------------------------------------------------------------------
+#-------------------------------------------------------------
+def get_fle_gauss(np):
+	"""
+	fle_gauss => the Legendre Polynomial function evaluated at gauss points
+			    	
+	"""
+	fle_ncb = [1.0]*(np+1)
+	fle_ncb = [fle0(gauss[0,0]),fle1(gauss[0,0]),fle2(gauss[0,0]),fle3(gauss[0,0])]
+
+	fle_pcb = [1.0]*(np+1)
+	fle_pcb = [fle0(gauss[1,0]),fle1(gauss[1,0]),fle2(gauss[1,0]),fle3(gauss[1,0])]
+
+	fle_nip = [1.0]*(np+1)
+	fle_nip = [fle0(gauss[2,0]),fle1(gauss[2,0]),fle2(gauss[2,0]),fle3(gauss[2,0])]
+
+	fle_pip = [1.0]*(np+1)
+	fle_pip = [fle0(gauss[3,0]),fle1(gauss[3,0]),fle2(gauss[3,0]),fle3(gauss[3,0])]
+
+
+#-------------------------------------------------------------------
+	fle_gauss = numpy.zeros((np+1,np+1), dtype=float)
+
+	for j in range(np+1):
+		fle_gauss[j,0] = fle_ncb[j]
+		fle_gauss[j,1] = fle_pcb[j]
+		fle_gauss[j,2] = fle_nip[j]
+		fle_gauss[j,3] = fle_pip[j]
+		
+	
+	return fle_gauss
+#-------------------------------------------------------------------
+#-------------------------------------------------------------------
+def get_fle_derivative(np):
+	""" fle_grad_gauss => the Legendre Polynomial derivative of function evaluated
+			   					 at gauss points 
+	"""
+
+	fle_grad_ncb = [1.0]*(np+1)
+	fle_grad_ncb = [fle0_grad(gauss[0,0]),fle1_grad(gauss[0,0]),fle2_grad(gauss[0,0]),fle3_grad(gauss[0,0])]
+
+	fle_grad_pcb = [1.0]*(np+1)
+	fle_grad_pcb = [fle0_grad(gauss[1,0]),fle1_grad(gauss[1,0]),fle2_grad(gauss[1,0]),fle3_grad(gauss[1,0])]
+
+	fle_grad_nip = [1.0]*(np+1)
+	fle_grad_nip = [fle0_grad(gauss[2,0]),fle1_grad(gauss[2,0]),fle2_grad(gauss[2,0]),fle3_grad(gauss[2,0])]
+
+	fle_grad_pip = [1.0]*(np+1)
+	fle_grad_pip = [fle0_grad(gauss[3,0]),fle1_grad(gauss[3,0]),fle2_grad(gauss[3,0]),fle3_grad(gauss[3,0])]
+
+#----------------------------------------------------------------------
+	fle_grad_gauss = numpy.zeros((np+1,np+1), dtype=float)
+
+	for j in range(np+1):
+		fle_grad_gauss[j,0] = fle_grad_ncb[j]
+		fle_grad_gauss[j,1] = fle_grad_pcb[j]
+		fle_grad_gauss[j,2] = fle_grad_nip[j]
+		fle_grad_gauss[j,3] = fle_grad_pip[j]
+		
+#-----------------------------------------------------------------------
+	return fle_grad_gauss
+
+  
\ No newline at end of file
diff --git a/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc
new file mode 100644
index 0000000..ec05475
Binary files /dev/null and b/Martinez_1D_Euler/Legendre_Polynomial_Function_np3.pyc differ
diff --git a/Martinez_1D_Euler/Plot_Functions2.py b/Martinez_1D_Euler/Plot_Functions2.py
new file mode 100644
index 0000000..b2036de
--- /dev/null
+++ b/Martinez_1D_Euler/Plot_Functions2.py
@@ -0,0 +1,232 @@
+
+# Legendre Polynomial Function for Plots
+
+import sympy
+import numpy
+from sympy.utilities.lambdify import lambdify
+
+#-------------------------------------------------------
+np = 3
+xp = sympy.symbols('xp')
+#-------------------------------------------------------
+
+
+#Quadrature Points
+
+gauss_plot = numpy.zeros((np), dtype=float)
+gauss_plot[0]=-1./3.
+gauss_plot[1]=0.
+gauss_plot[2]=1./3.
+
+#----------------------------------------------------------
+#Legendre Polynomial Terms
+fle = [1.0]*(np+1)
+
+fle[0] = 1.0
+fle[1] = xp
+fle[2] = (xp**2)-(1.0/12.0)
+fle[3] = (xp**3)-(0.15*xp)
+ 
+#---------------------------------------------------------
+fle_eval = [1.0]*(np+1)
+
+for i in range(np+1):
+	fle_eval[i] = lambdify((xp),fle[i])
+#----------------------------------------------------------
+fle_0_eval = fle_eval[0]
+fle_1_eval = fle_eval[1]
+fle_2_eval = fle_eval[2]
+fle_3_eval = fle_eval[3]
+
+#----------------------------------------------------------
+def fle0(xp):
+	f0 = fle_0_eval(xp)
+	return f0
+
+def fle1(xp):
+	f1 = fle_1_eval(xp)
+	return f1
+
+def fle2(xp):
+	f2 = fle_2_eval(xp)
+	return f2
+    
+def fle3(xp):
+	f3 = fle_3_eval(xp)
+	return f3
+
+#----------------------------------------------------------------------	
+
+
+fle_np = [1.0]*(np+1)
+fle_np = [fle0(gauss_plot[0]),fle1(gauss_plot[0]),fle2(gauss_plot[0]),fle3(gauss_plot[0])]
+
+fle_zero = [1.0]*(np+1)
+fle_zero = [fle0(gauss_plot[1]),fle1(gauss_plot[1]),fle2(gauss_plot[1]),fle3(gauss_plot[1])]
+
+fle_pp = [1.0]*(np+1)
+fle_pp = [fle0(gauss_plot[2]),fle1(gauss_plot[2]),fle2(gauss_plot[2]),fle3(gauss_plot[2])]
+
+#-------------------------------------------------------------------
+fle_gauss_plot = numpy.zeros((np+1,np), dtype=float)
+
+for j in range(np+1):
+	fle_gauss_plot[j,0] = fle_np[j]
+	fle_gauss_plot[j,1] = fle_zero[j]
+	fle_gauss_plot[j,2] = fle_pp[j]
+
+#-----------------------------------------------------------------------------		
+def plot_density2(ncell,np, Q, x, dx):
+	
+	
+	rho1 = numpy.zeros((ncell), dtype=float)
+	rho2 = numpy.zeros((ncell), dtype=float)
+	rho3 = numpy.zeros((ncell), dtype=float)
+	
+	xx1 = numpy.zeros((ncell), dtype=float)
+	xx2 = numpy.zeros((ncell), dtype=float)
+	xx3 = numpy.zeros((ncell), dtype=float)
+	
+	for i in range(ncell):
+		for j in range(np):
+			
+			rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i]
+			rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i]
+			rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i]
+			
+		xx1[i] = x[i] + (-1./3.)*dx[i]
+		xx2[i] = x[i] + (0.)*dx[i]
+		xx3[i] = x[i] + (1./3.)*dx[i]
+
+		
+	rho_t2 = numpy.concatenate((rho1,rho2, rho3))
+	xx_t2 = numpy.concatenate((xx1, xx2, xx3))
+		
+	return xx_t2, rho_t2
+	
+#-----------------------------------------------------------------------------		
+def plot_velocity2(ncell,np, Q, x, dx):
+
+	rho1 = numpy.zeros((ncell), dtype=float)
+	rho2 = numpy.zeros((ncell), dtype=float)
+	rho3 = numpy.zeros((ncell), dtype=float)
+	
+	mom1 = numpy.zeros((ncell), dtype=float)
+	mom2 = numpy.zeros((ncell), dtype=float)
+	mom3 = numpy.zeros((ncell), dtype=float)
+	
+	vel1 = numpy.zeros((ncell), dtype=float)
+	vel2 = numpy.zeros((ncell), dtype=float)
+	vel3 = numpy.zeros((ncell), dtype=float)
+	
+	xx1 = numpy.zeros((ncell), dtype=float)
+	xx2 = numpy.zeros((ncell), dtype=float)
+	xx3 = numpy.zeros((ncell), dtype=float)
+	
+	for i in range(ncell):
+		for j in range(np):
+			
+			rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i]
+			rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i]
+			rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i]
+			
+			mom1[i] += fle_gauss_plot[j,0]*Q[1,j,i]
+			mom2[i] += fle_gauss_plot[j,1]*Q[1,j,i]
+			mom3[i] += fle_gauss_plot[j,2]*Q[1,j,i]
+		
+		vel1[i] = mom1[i]/rho1[i]
+		vel2[i] = mom2[i]/rho2[i]
+		vel3[i] = mom3[i]/rho3[i]
+			
+		xx1[i] = x[i] + (-1./3.)*dx[i]
+		xx2[i] = x[i] + (0.)*dx[i]
+		xx3[i] = x[i] + (1./3.)*dx[i]
+		
+	vel_t2 = numpy.concatenate((vel1,vel2, vel3))
+	xx_t2 = numpy.concatenate((xx1, xx2, xx3))
+		
+	return xx_t2, vel_t2
+	
+#------------------PLOT Pressure!-----------------------------------------------	
+
+def plot_pressure2(ncell,np, Q, x, dx):
+	
+	gamma = 1.4
+	
+	rho1 = numpy.zeros((ncell), dtype=float)
+	rho2 = numpy.zeros((ncell), dtype=float)
+	rho3 = numpy.zeros((ncell), dtype=float)
+	
+	mom1 = numpy.zeros((ncell), dtype=float)
+	mom2 = numpy.zeros((ncell), dtype=float)
+	mom3 = numpy.zeros((ncell), dtype=float)
+	
+	en1 = numpy.zeros((ncell), dtype=float)
+	en2 = numpy.zeros((ncell), dtype=float)
+	en3 = numpy.zeros((ncell), dtype=float)
+	
+	press1 = numpy.zeros((ncell), dtype=float)
+	press2 = numpy.zeros((ncell), dtype=float)
+	press3 = numpy.zeros((ncell), dtype=float)
+		
+	xx1 = numpy.zeros((ncell), dtype=float)
+	xx2 = numpy.zeros((ncell), dtype=float)
+	xx3 = numpy.zeros((ncell), dtype=float)
+	
+	for i in range(ncell):
+		for j in range(np):
+			
+			rho1[i] += fle_gauss_plot[j,0]*Q[0,j,i]
+			rho2[i] += fle_gauss_plot[j,1]*Q[0,j,i]
+			rho3[i] += fle_gauss_plot[j,2]*Q[0,j,i]
+			
+			mom1[i] += fle_gauss_plot[j,0]*Q[1,j,i]
+			mom2[i] += fle_gauss_plot[j,1]*Q[1,j,i]
+			mom3[i] += fle_gauss_plot[j,2]*Q[1,j,i]
+			
+			en1[i] += fle_gauss_plot[j,0]*Q[2,j,i]
+			en2[i] += fle_gauss_plot[j,1]*Q[2,j,i]
+			en3[i] += fle_gauss_plot[j,2]*Q[2,j,i]
+			
+		press1[i] = (gamma-1.)*(en1[i]- 0.5*(mom1[i]**2)/rho1[i]) 
+		press2[i] = (gamma-1.)*(en2[i]- 0.5*(mom2[i]**2)/rho2[i]) 
+		press3[i] = (gamma-1.)*(en3[i]- 0.5*(mom3[i]**2)/rho3[i]) 
+			
+		xx1[i] = x[i] + (-1./3.)*dx[i]
+		xx2[i] = x[i] + (0.)*dx[i]
+		xx3[i] = x[i] + (1./3.)*dx[i]
+		
+	press_t2 = numpy.concatenate((press1,press2, press3))
+	xx_t2 = numpy.concatenate((xx1, xx2, xx3))
+		
+	return xx_t2, press_t2
+	
+#------------------PLOT Energy!-----------------------------------------------	
+
+def plot_energy2(ncell,np, Q, x, dx):	
+
+	en1 = numpy.zeros((ncell), dtype=float)
+	en2 = numpy.zeros((ncell), dtype=float)
+	en3 = numpy.zeros((ncell), dtype=float)
+	
+	xx1 = numpy.zeros((ncell), dtype=float)
+	xx2 = numpy.zeros((ncell), dtype=float)
+	xx3 = numpy.zeros((ncell), dtype=float)
+	
+	for i in range(ncell):
+		for j in range(np):
+			
+			en1[i] += fle_gauss_plot[j,0]*Q[2,j,i]
+			en2[i] += fle_gauss_plot[j,1]*Q[2,j,i]
+			en3[i] += fle_gauss_plot[j,2]*Q[2,j,i]
+	
+		xx1[i] = x[i] + (-1./3.)*dx[i]
+		xx2[i] = x[i] + (0.)*dx[i]
+		xx3[i] = x[i] + (1./3.)*dx[i]
+	
+	en_t2 = numpy.concatenate((en1,en2, en3))
+	xx_t2 = numpy.concatenate((xx1, xx2, xx3))
+		
+	return xx_t2, en_t2
+
+	
\ No newline at end of file
diff --git a/Martinez_1D_Euler/Plot_Functions2.pyc b/Martinez_1D_Euler/Plot_Functions2.pyc
new file mode 100644
index 0000000..a43dc9e
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diff --git a/Martinez_1D_Euler/Residual_Function_np3.py b/Martinez_1D_Euler/Residual_Function_np3.py
new file mode 100644
index 0000000..9b23cee
--- /dev/null
+++ b/Martinez_1D_Euler/Residual_Function_np3.py
@@ -0,0 +1,349 @@
+import numpy
+import math
+
+np = 3 # order of scheme
+
+# Store the spectral coefficients for np = 3 and a fifth order polynomial:
+volcoef = [0.0]*(np+1)
+
+volcoef[0] = 1.0/12.0
+volcoef[1] = 5.0/12.0
+volcoef[2] = 5.0/12.0
+volcoef[3] = 1.0/12.0
+
+ncell = 10 #number of cells
+nface = ncell+1 #cell boundaries
+rk3 = 3 #This is because we are solving 3 equations(mass, momentum, energy)
+
+Qf = numpy.zeros((rk3,np+1,ncell), dtype=float)
+
+Qfl = numpy.empty((rk3),dtype=float)
+Qfr = numpy.empty((rk3),dtype=float)
+
+Fl = numpy.empty((rk3),dtype=float)
+bflux = numpy.zeros((rk3, ncell, 2), dtype=float)
+
+volflux = numpy.zeros((rk3,np+1,ncell), dtype=float)
+h = numpy.zeros((rk3,np,ncell), dtype=float)
+
+#f2c = numpy.empty((nface, 2), dtype=float)
+
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#---------------------------------------------------------------
+
+def get_Qf(ncell, rk3, np, fle_gauss, Q):
+
+	#Here we approximate U(x,t) = sum (basis_function(x)* U(t))
+	
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				Qf[iv,0,i] += fle_gauss[k,0]*Q[iv,k,i]
+				Qf[iv,1,i] += fle_gauss[k,2]*Q[iv,k,i]
+				Qf[iv,2,i] += fle_gauss[k,3]*Q[iv,k,i]
+				Qf[iv,3,i] += fle_gauss[k,1]*Q[iv,k,i]
+				
+	return Qf
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#-----------BOUNDARY CONDITIONS---------------------------------
+
+def get_Qbright_Qbleft(rk3,np, ncell, Qf):
+	
+	#Apply Boundary Conditions
+	
+	# Try to change the boundary conditions to see the impact to the solver
+	#currently in comment block are the reflecting boundary conditions
+	
+
+	Qbleft = [0.0]*(rk3)
+	Qbright = [0.0]*(rk3)
+# 
+# 	Qbleft[0] = Qf[0,0,0]
+#  	Qbleft[1] = -Qf[1,0,0]
+#  	Qbleft[2] = Qf[2,0,0]
+#  
+#  	Qbright[0] = Qf[0,np, ncell-1]
+#  	Qbright[1] = -Qf[1, np, ncell-1]
+#  	Qbright[2] = Qf[2, np, ncell-1]
+	
+	Qbleft[0] = 1.25
+ 	Qbleft[1] = 0.0
+ 	Qbleft[2] = 2.5
+ 
+ 	Qbright[0] = 0.125
+ 	Qbright[1] = 0.0
+ 	Qbright[2] = 0.25
+	
+	return Qbleft, Qbright
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#------------------BC FOR 1st CELL LEFT EDGE---------------------
+
+def get_Qfl_Qfr_left(rk3,Qf,np, Qbleft):
+
+
+	for iv in range(rk3):
+		Qfl[iv] = Qbleft[iv]
+		Qfr[iv] = Qf[iv,0,0]
+
+	return Qfl, Qfr
+#---------------------------------------------------------------
+
+#Fl = get_rusanov_flux(Qfl, Qfr, normal, rk3)
+
+def get_bflux_left(Fl):
+	bflux[:,0,0] = Fl[:]
+	return bflux
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#------------------BC FOR LAST CELL RIGHT EDGE-------------------
+
+def get_Qfl_right(Qf,np, ncell):
+	
+	# Qfl = [1.0]*(rk3)
+	
+	Qfl[:] = Qf[:,np, ncell-1]
+	return Qfl
+	
+#---------------------------------------------------------------
+#Fl = get_rusanov_flux(Qfl, Qbright, normal, rk3)
+
+
+def get_bflux_right(rk3, ncell, Fl):
+	for i in range(rk3):
+		bflux[i,ncell-1,1] = Fl[i]
+	
+	return bflux
+	
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#------------------Interior Edge Flux----------------------------
+
+def get_bflux_interior(nface, f2c, Qf, np, get_rusanov_flux):
+	
+	normal = 1.0
+	
+	ic_left = numpy.empty((nface-1), dtype=float)
+	ic_right = numpy.empty((nface-1), dtype=float)
+	Fl = numpy.empty((rk3), dtype = float)
+	
+	for i in range(1,nface-1):
+		for j in range(rk3):
+			
+			ic_left[i] = f2c[i,0]
+			ic_right[i] = f2c[i,1]
+			
+			ic_l = ic_left[i]
+			ic_r = ic_right[i]
+    
+        	Qfl[j] = Qf[j, np, ic_l] 
+        	Qfr[j] = Qf[j,0, ic_r]
+		
+		#----------------------------------#
+		#Incorporate Rusanov flux function:
+		
+		"Returns the fluxes"
+		gamma = 1.4
+	
+		rhol = numpy.abs(Qfl[0])
+		ul   = Qfl[1]/rhol
+		pl   = (gamma-1.0)*(Qfl[2]-(0.5*rhol*(ul**2)))
+		
+		rhor = numpy.abs(Qfr[0])
+		ur   = Qfr[1]/rhor
+		pr   = (gamma-1.0)*(Qfr[2]-(0.5*rhor*(ur**2)))
+
+		
+		c_a  = numpy.sqrt(((gamma*numpy.abs(pl+pr))/(rhol+rhor)))
+		eigv = 0.5*(numpy.absolute(ul+ur)) + c_a 
+
+		
+		"Calculate the flux components"
+	
+		Fl[0] = Qfl[0]*ul + Qfr[0]*ur
+		Fl[1] = Qfl[1]*ul + Qfr[1]*ur + (pl+pr)
+		Fl[2] = (Qfl[2]+pl)*ul + (Qfr[2]+pr)*ur
+		
+		for iv in range(rk3):
+			Fl[iv] = (Fl[iv]*normal - (Qfr[iv]-Qfl[iv]))*eigv*0.5
+		
+		#---------------------------------------#
+		for j in range(rk3):
+			bflux[j,ic_l,1]  = Fl[j]
+			bflux[j,ic_r,0] = Fl[j]
+    
+	return bflux
+	
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#----------------Compute All Volume Fluxes----------------------
+
+
+
+
+def get_volflux(rk3, np, ncell, Qf):
+	
+	volflux = numpy.zeros((rk3,np+1,ncell), dtype=float)
+	Fl = numpy.empty((rk3), dtype = float)
+	
+	for i in range(ncell):
+		for j in range(np+1):
+			for iv in range(rk3):
+				Qfl[iv] = Qf[iv,j,i]
+				
+				gamma = 1.4
+				rhol = numpy.abs(Qfl[0])
+				ul   = Qfl[1]/rhol
+				pl   = (gamma-1.0)*(Qfl[2]-(0.5*Qfl[0]*(ul**2)))
+				
+				Fl[0] = Qfl[0]*ul 
+				Fl[1] = Qfl[1]*ul + pl
+				Fl[2] = (Qfl[2]+pl)*ul
+				
+				
+			for iv in range(rk3):
+				volflux[iv,j,i] = Fl[iv]
+	
+	return volflux
+        		
+
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#-----------Get Initial h Matrix (RHS)--------------------------
+
+def get_initial_hmatrix(rk3, np, ncell, bflux, fle_gauss):
+
+	bl = [0.0]*(np)
+	br = [0.0]*(np)
+
+	for k in range(np):
+		bl[k] = fle_gauss[k,0]
+		br[k] = fle_gauss[k,1] 
+		
+	h = numpy.zeros((rk3,np,ncell), dtype=float)
+
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				h[iv,k,i] = (bflux[iv,i,0]*bl[k]) - (bflux[iv,i,1]*br[k])
+	return h
+	
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+
+#--------------Volume Integral by Gauss Lobatto----------------
+
+def get_GL_hmatrix(rk3, np, ncell, h, volflux, volcoef, fle_grad_gauss):
+	for i in range(ncell):
+		for iv in range(rk3):
+			for j in range(1, np):
+				h[iv,j,i] = (h[iv,j,i] + 
+                         	volflux[iv,0,i]*volcoef[0]*fle_grad_gauss[j,0] +
+                         	volflux[iv,1,i]*volcoef[1]*fle_grad_gauss[j,2] +
+                         	volflux[iv,2,i]*volcoef[2]*fle_grad_gauss[j,3] + 
+                         	volflux[iv,3,i]*volcoef[3]*fle_grad_gauss[j,1] )
+                         	
+	return h 
+		
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------	
+
+#-------------Compute the Mass Matrix --------------------------
+
+def get_residual_matrix(rk3, np, ncell, aic, h, dx):
+
+	residual = numpy.zeros((rk3,np,ncell))
+	
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				residual[iv,k,i] = aic[k,k]*h[iv,k,i]/dx[i]
+				
+	return residual
+	
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+#---------------------------------------------------------------
+		
+def get_rusanov_flux(Ql, Qr, unitnorm, rk3):
+	
+	"Returns the fluxes"
+	gamma = 1.4
+	
+	rhol = numpy.abs(Ql[0])
+	ul   = Ql[1]/rhol
+	pl   = (gamma-1.0)*(Ql[2]-(0.5*rhol*(ul**2)))
+		
+	rhor = numpy.abs(Qr[0])
+	ur   = Qr[1]/rhor
+	pr   = (gamma-1.0)*(Qr[2]-(0.5*rhor*(ur**2)))
+		
+	c_a  = numpy.sqrt(((gamma*numpy.abs(pl+pr))/(rhol+rhor)))
+	eigv = 0.5*(numpy.absolute(ul+ur)) + c_a 
+
+		
+	"Calculate the flux components"
+	
+	Fl = numpy.empty((rk3), dtype = float)
+	
+	Fl[0] = Ql[0]*ul + Qr[0]*ur
+	Fl[1] = Ql[1]*ul + Qr[1]*ur + (pl+pr)
+	Fl[2] = (Ql[2]+pl)*ul + (Qr[2]+pr)*ur
+		
+	for iv in range(rk3):
+		Fl[iv] = (Fl[iv]*unitnorm - (Qr[iv]-Ql[iv]))*eigv*0.5
+	
+	return Fl
+#----------------------------------------------------------------
+#----------------------------------------------------------------
+#----------------------------------------------------------------
+#----------------------------------------------------------------
+
+#All volume flux:
+
+def get_volume_flux(Ql):
+
+	"Returns the volume fluxes"
+	
+	gamma = 1.4
+	rhol = numpy.abs(Ql[0])
+	ul   = Ql[1]/rhol
+	pl   = (gamma-1.0)*(Ql[2]-(0.5*Ql[0]*(ul**2)))
+	
+	"Calculate the flux components"
+	Fl[0] = numpy.abs(Ql[0]*ul) 
+	Fl[1] = Ql[1]*ul + pl
+	Fl[2] = (Ql[2]+pl)*ul 
+	
+	return Fl
+#--------------------------------------------------------------------
+
diff --git a/Martinez_1D_Euler/Residual_Function_np3.pyc b/Martinez_1D_Euler/Residual_Function_np3.pyc
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diff --git a/Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.py b/Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.py
new file mode 100644
index 0000000..91438a8
--- /dev/null
+++ b/Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.py
@@ -0,0 +1,53 @@
+import numpy          
+#---------------------------------------------------
+# ----------------Stage 1 Runge Kutta---------------
+
+def get_Qrk_stage1(Q_rk, ncell, rk3, np, dt, residual):
+	
+	Qrk1 = numpy.zeros((rk3,np, ncell), dtype=float)
+	
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				Qrk1[iv,k,i] = Q_rk[iv,k,i]+(dt*residual[iv,k,i])
+				
+	return Qrk1
+            
+#------------------------------------------------------
+#------------------------------------------------------
+# ----------------Stage 1 Runge Kutta------------------
+
+def get_Qrk_stage2(Q_rk, Qrk1, ncell, rk3, np, dt, residual):
+
+	Qrk2 = numpy.zeros((rk3,np, ncell), dtype=float)
+	
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				Qrk2[iv,k,i] = (0.75*Q_rk[iv,k,i])+0.25*(Qrk1[iv,k,i]+(dt*residual[iv,k,i]))
+				
+	return Qrk2
+            
+#------------------------------------------------------
+#------------------------------------------------------
+# -----------------Stage 3 Runge Kutta-----------------
+
+def get_Qrk_stage3(Q_rk, Qrk2, ncell, rk3, np, dt, residual):
+
+	Qrk3 = numpy.zeros((rk3,np, ncell), dtype=float)
+	
+	for i in range(ncell):
+		for iv in range(rk3):
+			for k in range(np):
+				Qrk3[iv,k,i] = ((1.0/3.0)*(Q_rk[iv,k,i])) + ((2.0/3.0)*(Qrk2[iv,k,i]+(dt*residual[iv,k,i])))
+	
+	return Qrk3
+#------------------------------------------------------           
+#ctime = ctime + 0.5*dt
+#------------------------------------------------------
+'''residnorm = 0.0
+
+for i in range(ncell):
+    for k in range(np):
+        residnorm = residnorm + numpy.abs(residual[2,k,i])
+        '''
diff --git a/Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.pyc b/Martinez_1D_Euler/Runge_Kutta_Third_Order_np3.pyc
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diff --git a/Martinez_1D_Euler/U_initial_np3.py b/Martinez_1D_Euler/U_initial_np3.py
new file mode 100644
index 0000000..a56b328
--- /dev/null
+++ b/Martinez_1D_Euler/U_initial_np3.py
@@ -0,0 +1,83 @@
+import numpy
+
+np = 3
+
+aic = numpy.zeros((np, np), dtype=float)
+
+
+# Store the spectral coefficients for np = 3 and a fifth order polynomial:
+volcoef = [0.0]*(np+1)
+
+volcoef[0] = 1.0/12.0
+volcoef[1] = 5.0/12.0
+volcoef[2] = 5.0/12.0
+volcoef[3] = 1.0/12.0
+
+# Store the Quadrature Points and Weights for 4 point quadrature(np=3), fifth order polynomial:
+
+#Quadrature Points
+
+gauss = numpy.zeros((np+1, np+1), dtype=float)
+gauss[0,0]=-0.5
+gauss[1,0]=0.5
+gauss[2,0]= -1 * numpy.sqrt(5)/10
+gauss[3,0]= numpy.sqrt(5)/10.0
+
+# Corresponding Quadrature weights
+
+gauss[0,1]=1.0/12.0
+gauss[1,1]=gauss[0,1]
+gauss[2,1]=5.0/12.0
+gauss[3,1]=gauss[2,1]
+
+#Inverse Mass Matrix
+aic_3 = numpy.zeros((np,np), dtype=float)
+aic_3[0,0]=1.0
+aic_3[1,1]=12.0
+aic_3[2,2]=180.0
+
+#---------------------------------------
+"""Defines inverse mass matrix and GL x points and weights
+"""
+def get_inverse_mass_matrix(np):
+	
+	#Third Order:Store the Inverse of the Mass Matrix
+	
+	for i in range(np):
+		for j in range(np):
+			aic[i,j] = aic_3[i,j]
+	
+	return aic
+#------------------------------------------------------------------
+
+def get_U_initial(np, rk3, ncell, fle_gauss, x , dx, aic):
+	block = 0.50
+	
+	qinit = numpy.zeros((np+1,rk3,ncell), dtype=float)
+	U_initial = numpy.empty((rk3,np,ncell), dtype=float)
+	atemp = numpy.empty((np,rk3), dtype=float)
+	
+	for i in range(ncell):
+		for j in range(np):
+			for k in range(3):
+			
+				xeval = x[i] + gauss[j,0] * dx[i]
+				if (xeval < block):
+					qinit[k,0,i] = 1.25
+					qinit[k,1,i] = 0.0
+					qinit[k,2,i] = 2.5
+				
+				if (xeval >= block):
+					qinit[k,0,i] = 0.125
+					qinit[k,1,i] = 0.0
+					qinit[k,2,i] = 0.25
+					
+				atemp[k,j] = ((qinit[0,k,i]*fle_gauss[j,0])*gauss[0,1] + (qinit[1,k,i]*fle_gauss[j,1])*gauss[1,1] +
+                          	   (qinit[2,k,i]*fle_gauss[j,2])*gauss[2,1] + (qinit[3,k,i]*fle_gauss[j,3])*gauss[3,1])
+
+			U_initial[0,j,i]=aic[j,j]*atemp[0,j]
+			U_initial[1,j,i]=aic[j,j]*atemp[1,j]
+			U_initial[2,j,i]=aic[j,j]*atemp[2,j]
+				
+	return U_initial
+#-----------------------------------------------------------------------
diff --git a/Martinez_1D_Euler/U_initial_np3.pyc b/Martinez_1D_Euler/U_initial_np3.pyc
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diff --git a/Martinez_1D_Euler/meshing.py b/Martinez_1D_Euler/meshing.py
new file mode 100644
index 0000000..067b313
--- /dev/null
+++ b/Martinez_1D_Euler/meshing.py
@@ -0,0 +1,47 @@
+#Specify the number of faces for flux
+import numpy
+ncell = 10	
+#------------------------------------
+# set up the initial condition
+xleft=  0.0
+xlen= 1.0
+xright= xlen + xleft
+dxuni=xlen/ncell
+dxmin=1.0
+#------------------------------------
+temp = numpy.empty((ncell+1), dtype=float)
+for i in range(ncell+1):
+    temp[i] = dxuni * (i)
+       
+#------------------------------------
+def get_x_cell(ncell):
+	x = numpy.empty((ncell), dtype=float)
+	
+	for i in range(ncell):
+		x[i]=(temp[i+1]+(temp[i]))/2.0
+	
+	return x
+	
+#---------------------------------------
+
+def get_dx_cell(ncell):
+	dx = numpy.empty((ncell), dtype=float)
+	
+	for i in range(ncell):
+		dx[i]=temp[i+1]-temp[i]
+		
+	return dx
+	
+#-----------------------------------------
+
+def get_face_to_cell(nface):
+	
+	f2c = numpy.empty((nface, 2), dtype=float)
+	for i in range(nface):
+		f2c[i,0] = i-1
+		f2c[i,1] = i
+	f2c[nface-1,1] = 0.0
+	f2c[0,0] = 0.0
+	
+	return f2c
+
diff --git a/Martinez_1D_Euler/meshing.pyc b/Martinez_1D_Euler/meshing.pyc
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