WallGo.grid.Grid

class Grid(M, N, positionFalloff, momentumFalloffT, spacing='Spectral')

Bases: object

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’.

__init__(M, N, positionFalloff, momentumFalloffT, spacing='Spectral')

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’.

Methods

__init__(M, N, positionFalloff, momentumFalloffT)	Initialises Grid object.
changeMomentumFalloffScale(newScale)	Change the momentum falloff scale.
changePositionFalloffScale(newScale)	Change the position falloff scale.
compactificationDerivatives(zCompact, ...)	Derivative \(d(X)/d(X_\text{compact})\) of coordinate transforms to [-1, 1] interval.
compactify(z, pz, pp)	Transforms coordinates to [-1, 1] interval
decompactify(zCompact, pzCompact, ppCompact)	Transforms coordinates to [-1, 1] interval
getCompactCoordinates([endpoints, direction])	Return compact coordinates of grid.
getCompactificationDerivatives([endpoints])	Return derivatives of compactified coordinates of grid, with respect to uncompactified derivatives.
getCoordinates([endpoints])	Return coordinates of grid, not compactified.

changeMomentumFalloffScale(newScale)

Change the momentum falloff scale.

Parameters:

newScale (float) – New momentum falloff scale.

Return type:

None

changePositionFalloffScale(newScale)

Change the position falloff scale.

Parameters:

newScale (float) – New position falloff scale.

Return type:

None

compactificationDerivatives(zCompact, pzCompact, ppCompact)

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)

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)

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)

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)

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)

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, …]