codeNote.md

Coding note

1d h-Based Richards equation

$$ C(h) \frac{\partial h }{\partial t}- \frac{\partial }{\partial z}[k\frac{\partial h }{\partial z}] +\frac{\partial k }{\partial z}=0 $$ where $C$, $k$ are nonlinear function of $h$

finite difference approximation

Using a 1st order Taylor expansion

a Picard iteration FDM scheme

$$ C_i^{n+1,m} \times \frac{ h_i^{n+1,m}- h_i^n }{\Delta t}- [\frac{ k^{n+1,m}_{i+1/2}(h_{i+1}^{n+1,m+1}-h_i^{n+1,m+1}) -k^{n+1,m}_{i-1/2}(h_{i}^{n+1,m+1}-h_{i-1}^{n+1,m+1})} { \Delta z^2 }] \\ +\frac{k^{n+1,m}_{i+1/2}-k^{n+1,m}_{i-1/2}}{\Delta z} $$

boundary condition

for inner nodes i (where neighbours are free nodes)

$$ w_{i-1}h_{i-1}+w_{i}h_{i}+w_{i-1}h_{i-1}=b_i $$ where the weight could be calculated from above equation easily.

Assemble

we now have linear equation to be solve at each iteration m $Ah=b$. A is a sparse band matrix. Let this describes all node including those the values are known.

Define a picking up matrix $P$ that would pick up the row corresponding to free node.

we have $\hat{P}+P=I$

$$ (\hat{P}+P)Ah=b $$

$$ PAh=b-\hat{P}Ah $$ Sine the zero column of $P/\hat{P}$