WallGo.grid
Class for computing and storing the coordinates on the grid and other related quantities.
Classes
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Computes the grid on which the Boltzmann equation is solved. |
- class Grid(M, N, positionFalloff, momentumFalloffT, spacing='Spectral')[source]
Computes the grid on which the Boltzmann equation is solved.
Grid is 3d, and consists of the physical coordinates:
\(\xi\), position perpendicular to the wall,
\(p_z\), momentum perpendicular to the wall,
\(p_\Vert\), momentum magnitude parallel to the wall.
In addition there are the corresponding compactified coordinates on the interval [-1, 1],
\[\chi \equiv \frac{\xi}{\sqrt{\xi^2 + L_{\xi}^2}}, \qquad \rho_{z} \equiv \tanh\left(\frac{p_z}{2 T_0}\right), \qquad \rho_{\Vert} \equiv 1 - 2 e^{-p_\Vert/T_0}.\]All coordinates are in the wall frame.
- chiValues
Grid of the \(\chi\) direction.
- Type:
array_like
- rzValues
Grid of the \(\rho_z\) direction.
- Type:
array_like
- rpValues
Grid of the \(\rho_\Vert\) direction.
- Type:
array_like
- xiValues
Grid of the \(\xi\) direction.
- Type:
array_like
- pzValues
Grid of the \(p_z\) direction.
- Type:
array_like
- ppValues
Grid of the \(p_\Vert\) direction.
- Type:
array_like
Initialises Grid object.
Compactified coordinates are chosen according to
\[\chi = -\cos\left(\frac{\pi \alpha}{M}\right), \qquad \rho_{z} = -\cos\left(\frac{\pi \beta}{N}\right), \qquad \rho_{\Vert} = -\cos\left(\frac{\pi \gamma}{N-1}\right),\]with integers \(\alpha, \beta, \gamma\) taken over
\[\alpha = 0, 1, \dots, M, \qquad \beta = 0, 1, \dots, N, \qquad \gamma = 0, 1, \dots, N-1.\]These are the Gauss-Lobatto collocation points, here with all boundary points included.
The boundary points \(\chi=\pm 1\), \(\rho_z=\pm 1\) and \(\rho_{\Vert}=1\) correspond to points at infinity. The deviation from equilibrium is assumed to equal zero at infinity, so these points are dropped when solving the Boltzmann equations. The resulting grid is
\[\alpha = 1, 2, \dots, M-1, \qquad \beta = 1, 2, \dots, N-1, \qquad \gamma = 0, 1, \dots, N-2.\]- Parameters:
M (int) – Number of basis functions in the \(\xi\) (and \(\chi\)) direction.
N (int) – Number of basis functions in the \(p_z\) and \(p_\Vert\) (and \(\rho_z\) and \(\rho_\Vert\)) directions.
positionFalloff (float) – Length scale determining transform in \(\xi\) direction. Should be expressed in physical units (the units used in EffectivePotential).
momentumFalloffT (float) – Temperature scale determining transform in momentum directions. Should be close to the plasma temperature.
spacing ({'Spectral', 'Uniform'}) – Choose ‘Spectral’ for the Gauss-Lobatto collocation points, as required for WallGo’s spectral representation, or ‘Uniform’ for a uniform grid. Default is ‘Spectral’.
- changeMomentumFalloffScale(newScale)[source]
Change the momentum falloff scale.
- Parameters:
newScale (float) – New momentum falloff scale.
- Return type:
None
- changePositionFalloffScale(newScale)[source]
Change the position falloff scale.
- Parameters:
newScale (float) – New position falloff scale.
- Return type:
None
- compactificationDerivatives(zCompact, pzCompact, ppCompact)[source]
Derivative \(d(X)/d(X_\text{compact})\) of coordinate transforms to [-1, 1] interval.
- Parameters:
z_compact (array-like) – Compact z coordinate (chi).
pz_compact (array-like) – Compact pz coordinate (rho_z).
pp_compact (array-like) – Compact p_par coordinate (rho_par).
zCompact (ndarray)
pzCompact (ndarray)
ppCompact (ndarray)
- Returns:
dzdzCompact (array-like) – Derivative d(z)/d(chi).
dpzdpzCompact (array-like) – PDerivative d(p_z)/d(rho_z).
dppdppCompact (array-like) – Derivative d(p_par)/d(rho_par).
- Return type:
tuple[ndarray, …]
- compactify(z, pz, pp)[source]
Transforms coordinates to [-1, 1] interval
- Parameters:
z (array-like) – Physical z (or xi) coordinate.
pz (array-like) – Physical pz coordinate.
pp (array-like) – Physical p_par coordinate.
- Returns:
z_compact (array-like) – Compact z coordinate (chi).
pz_compact (array-like) – Compact pz coordinate (rho_z).
pp_compact (array-like) – Compact p_par coordinate (rho_par).
- Return type:
tuple[ndarray, …]
- decompactify(zCompact, pzCompact, ppCompact)[source]
Transforms coordinates to [-1, 1] interval
- Parameters:
z_compact (array-like) – Compact z coordinate (chi).
pz_compact (array-like) – Compact pz coordinate (rho_z).
pp_compact (array-like) – Compact p_par coordinate (rho_par).
zCompact (ndarray)
pzCompact (ndarray)
ppCompact (ndarray)
- Returns:
z (array-like) – Physical z (or xi) coordinate.
pz (array-like) – Physical pz coordinate.
pp (array-like) – Physical p_par coordinate.
- Return type:
tuple[ndarray, …]
- getCompactCoordinates(endpoints=False, direction=None)[source]
Return compact coordinates of grid.
- Parameters:
endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.
direction (string or None, optional) – Specifies which coordinates to return. Can either be ‘z’, ‘pz’, ‘pp’ or None. If None, returns a tuple containing the 3 directions. Default is None.
- Returns:
chiValues (array_like) – Grid of the \(\chi\) direction.
rzValues (array_like) – Grid of the \(\rho_z\) direction.
rpValues (array_like) – Grid of the \(\rho_\Vert\) direction.
- Return type:
tuple[ndarray, …] | ndarray
- getCompactificationDerivatives(endpoints=False)[source]
Return derivatives of compactified coordinates of grid, with respect to uncompactified derivatives.
- Parameters:
endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.
- Returns:
dchiValues (array_like) – Grid of the \(\partial_\xi\chi\) direction.
drzValues (array_like) – Grid of the \(\partial_{p_z}\rho_z\) direction.
drpValues (array_like) – Grid of the \(\partial_{p_\Vert}\rho_\Vert\) direction.
- Return type:
tuple[ndarray, …]
- getCoordinates(endpoints=False)[source]
Return coordinates of grid, not compactified.
- Parameters:
endpoints (Bool, optional) – If True, include endpoints of grid. Default is False.
- Returns:
xiValues (array_like) – Grid of the \(\xi\) direction.
pzValues (array_like) – Grid of the \(p_z\) direction.
ppValues (array_like) – Grid of the \(p_\Vert\) direction.
- Return type:
tuple[ndarray, …]