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minor tweaks to high order
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‎docs/content/3-code/6-higherorder.md‎

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@@ -34,36 +34,216 @@ The stencils are optimized for a given `CFL`, so it is important to use the one
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Note that the `CFL` is given in the standard convention! In order to use it with entity you need to multiply it by $\sqrt{N_\mathrm{dim}}$.
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| Solver | `CFL` | `delta_x` | `delta_y` | `delta_z` | `beta_xy` | `beta_yx` | `beta_xz` | `beta_zx` | `beta_yz` | `beta_zy` |
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| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
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| **2D** | | | | | | | | | | |
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| Yee | $1/\sqrt{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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| Cowan | 0.999 | 0 | 0 | 0 | 0.125 | 0.125 | 0 | 0 | 0 | 0 |
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| Lehe | 0.96 | -0.021 | 0 | 0 | 0.125 | 0.125 | 0 | 0 | 0 | 0 |
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| min1 | $0.97/\sqrt{2}$ | -0.125 | -0.125 | 0 | 0.11 | 0.11 | 0 | 0 | 0 | 0 |
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| min2 | $0.95/\sqrt{2}$ | -0.013 | -0.013 | 0 | -0.013 | -0.013 | 0 | 0 | 0 | 0 |
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| min3 | 0.5 | -0.065 | -0.065 | 0 | -0.065 | -0.065 | 0 | 0 | 0 | 0 |
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| min4 | 0.1 | -0.125 | -0.125 | 0 | -0.125 | -0.125 | 0 | 0 | 0 | 0 |
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| min5 | 0.96 | -0.017 | -0.017 | 0 | 0.133 | 0.133 | 0 | 0 | 0 | 0 |
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| min6 | 0.999 | -0.0005 | 0 | 0 | 0.128 | 0.128 | 0 | 0 | 0 | 0 |
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| **3D** | | | | | | | | | | |
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| Yee | $1/\sqrt{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
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| min3 | 0.5 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 |
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| min4 | 0.1 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 |
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<table id="blinne">
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<tr>
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<th>
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Dimension:
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</th>
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<th class="solvers-2d" colspan="9" style="text-align:center">
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2D
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</th>
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<th class="solvers-3d" colspan="3" style="text-align:center">
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3D
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</th>
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</tr>
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<tr>
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<th> Solver: </th>
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<th> Yee </th>
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<th> Cowan </th>
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<th> Lehe </th>
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<th> min1 </th>
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<th> min2 </th>
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<th> min3 </th>
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<th> min4 </th>
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<th> min5 </th>
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<th> min6 </th>
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<th> Yee </th>
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<th> min3 </th>
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<th> min4 </th>
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</tr>
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<tr>
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<th> <code>CFL</code> </th>
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<td> $1/\sqrt{2}$ </td>
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<td> $0.999$ </td>
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<td> $0.96$ </td>
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<td> $0.97/\sqrt{2}$ </td>
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<td> $0.95/\sqrt{2}$ </td>
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<td> $0.5$ </td>
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<td> $0.1$ </td>
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<td> $0.96$ </td>
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<td> $0.999$ </td>
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<td> $1/\sqrt{3}$ </td>
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<td> $0.5$ </td>
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<td> $0.1$ </td>
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</tr>
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<tr>
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<th> <code>delta_x</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.021$ </td>
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<td> $-0.125$ </td>
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<td> $-0.013$ </td>
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<td> $-0.065$ </td>
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<td> $-0.125$ </td>
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<td> $-0.017$ </td>
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<td> $-0.005$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>delta_y</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.125$ </td>
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<td> $-0.013$ </td>
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<td> $-0.065$ </td>
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<td> $-0.125$ </td>
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<td> $-0.017$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>delta_z</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_xy</code> </th>
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<td> $0$ </td>
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<td> $0.125$ </td>
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<td> $0.125$ </td>
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<td> $0.11$ </td>
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<td> $-0.013$ </td>
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<td> $-0.065$ </td>
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<td> $-0.125$ </td>
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<td> $0.133$ </td>
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<td> $0.128$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_yx</code> </th>
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<td> $0$ </td>
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<td> $0.125$ </td>
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<td> $0.125$ </td>
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<td> $0.11$ </td>
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<td> $-0.013$ </td>
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<td> $-0.065$ </td>
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<td> $-0.125$ </td>
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<td> $0.133$ </td>
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<td> $0.128$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_xz</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_zx</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_yz</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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<tr>
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<th> <code>beta_zy</code> </th>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $0$ </td>
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<td> $-0.00867$ </td>
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<td> $-0.048434$ </td>
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</tr>
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</table>
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<!-- | Solver | `CFL` | `delta_x` | `delta_y` | `delta_z` | `beta_xy` | `beta_yx` | `beta_xz` | `beta_zx` | `beta_yz` | `beta_zy` | -->
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<!-- | :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | -->
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<!-- | **2D** | | | | | | | | | | | -->
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<!-- | Yee | $1/\sqrt{2}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -->
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<!-- | Cowan | 0.999 | 0 | 0 | 0 | 0.125 | 0.125 | 0 | 0 | 0 | 0 | -->
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<!-- | Lehe | 0.96 | -0.021 | 0 | 0 | 0.125 | 0.125 | 0 | 0 | 0 | 0 | -->
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<!-- | min1 | $0.97/\sqrt{2}$ | -0.125 | -0.125 | 0 | 0.11 | 0.11 | 0 | 0 | 0 | 0 | -->
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<!-- | min2 | $0.95/\sqrt{2}$ | -0.013 | -0.013 | 0 | -0.013 | -0.013 | 0 | 0 | 0 | 0 | -->
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<!-- | min3 | 0.5 | -0.065 | -0.065 | 0 | -0.065 | -0.065 | 0 | 0 | 0 | 0 | -->
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<!-- | min4 | 0.1 | -0.125 | -0.125 | 0 | -0.125 | -0.125 | 0 | 0 | 0 | 0 | -->
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<!-- | min5 | 0.96 | -0.017 | -0.017 | 0 | 0.133 | 0.133 | 0 | 0 | 0 | 0 | -->
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<!-- | min6 | 0.999 | -0.0005 | 0 | 0 | 0.128 | 0.128 | 0 | 0 | 0 | 0 | -->
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<!-- | **3D** | | | | | | | | | | | -->
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<!-- | Yee | $1/\sqrt{3}$ | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | -->
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<!-- | min3 | 0.5 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -0.00867 | -->
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<!-- | min4 | 0.1 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -0.048434 | -->
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To visualize the impact of the field stencil you can see the field of a single particle traveling in vacuum at `gamma = 10.0` in the following plot.
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The rows show the particle traveling at $\theta = 0^\circ$, $\theta = 45^\circ$, $\theta = 90^\circ$ with respect to the x-axis.
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Each column corresponds to a different field stencil from the table above.
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![higherorder_stencil](../../assets/images/higherorder/fieldstencil_dark.png){width=100%, align=center}
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![higherorder_stencil](../../assets/images/higherorder/fieldstencil_dark.png#only-dark){width=100% align=center}
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![higherorder_stencil](../../assets/images/higherorder/fieldstencil_light.png#only-light){width=100% align=center}
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## Higher order shape functions
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To interpolate the current of a particle to the grid and the fields back to the particle, Entity uses a shape function. Before v1.3.0 this shape function was only of first order.
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Since v1.3.0 you can use shape functions with up-to 11th order and deposit charges with the scheme introduced by [Esirkepov (2001)](https://ui.adsabs.harvard.edu/abs/2001CoPhC.135..144E/abstract).
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![higherorder_shape](../../assets/images/higherorder/shape_functions_dark.png){width=50%, align=right}
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![higherorder_shape](../../assets/images/higherorder/shape_functions_dark.png){width=50% align=right class="invertdark"}
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You can switch to higher order shape functions by adding the follwoing compile flags to you `cmake` command: `-D deposit=esirkepov -D shape_order=<N>`, where `<N>` can be any number between `1` and `11`.
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From the top to bottom we uncrease the resolution of the grid to improve resolving the Debye-length.
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You can see that with higher order shape functions the numerical heating stops at considerably lower resolution.
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![higherorder_shape](../../assets/images/higherorder/heating_dark.png){width=100%, align=center}
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![higherorder_shape](../../assets/images/higherorder/heating_dark.png){width=100% align=center class="invertdark"}
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This allows you to reduce the numerical resolution when moving to higher order shape functions.
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We strongly advise to perform careful convergence tests before reducing numerical resolution in favor of higher order shape functions.
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The computational cost increase is negligible for 1D, but can become substantial for 3D.
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You can find scaling tests performed on a single Intel PVC GPU in the following plot.
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![higherorder_shape](../../assets/images/higherorder/scaling_dark.png){width=100%, align=center}
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![higherorder_shape](../../assets/images/higherorder/scaling_dark.png#only-dark){width=100% align=center}
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![higherorder_shape](../../assets/images/higherorder/scaling_light.png#only-light){width=100% align=center}

‎sass/components/_image.scss‎

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@use "../colors" as colors;
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img.invertdark {
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@include colors.light-theme {
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filter: invert(1) hue-rotate(180deg);
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}
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}

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