4.1.2.1.4 pde.backends.numba.operators.polar_sym module

This module implements differential operators on polar grids.

make_laplace	Make a discretized laplace operator for a polar grid.
make_gradient	Make a discretized gradient operator for a polar grid.
make_gradient_squared	Make a discretized gradient squared operator for a polar grid.
make_divergence	Make a discretized divergence operator for a polar grid.
make_vector_gradient	Make a discretized vector gradient operator for a polar grid.
make_tensor_divergence	Make a discretized tensor divergence operator for a polar grid.

make_divergence(grid, *, backend=None)

Make a discretized divergence operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType

make_gradient(grid, *, backend=None, method='central')

Make a discretized gradient operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

• method (str) – The method for calculating the derivative. Possible values are ‘central’, ‘forward’, and ‘backward’.

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType

make_gradient_squared(grid, *, backend=None, central=True)

Make a discretized gradient squared operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

• central (bool) – Whether a central difference approximation is used for the gradient operator. If this is False, the squared gradient is calculated as the mean of the squared values of the forward and backward derivatives.

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType

make_laplace(grid, *, backend=None)

Make a discretized laplace operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType

make_tensor_divergence(grid, *, backend=None)

Make a discretized tensor divergence operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType

make_vector_gradient(grid, *, backend=None)

Make a discretized vector gradient operator for a polar grid.

The polar grid assumes polar symmetry, so that fields only depend on the radial coordinate r. The radial discretization is defined as \(r_i = r_\mathrm{min} + (i + \frac12) \Delta r\) for \(i=0, \ldots, N_r-1\), where \(r_\mathrm{min}\) is the radius of the inner boundary, which is zero by default. Note that the radius of the outer boundary is given by \(r_\mathrm{max} = r_\mathrm{min} + N_r \Delta r\).

Parameters:

• grid (PolarSymGrid) – The polar grid for which this operator will be defined

• backend (NumbaBackend) – References to the backend to read configuration details

Returns:

A function that can be applied to an array of values

Return type:

OperatorImplType