`pymor.discretizers.builtin.fv`¶

This module provides some operators for finite volume discretizations.

Module Contents¶

class pymor.discretizers.builtin.fv.BoundaryL2Functional(grid, function, boundary_type=None, boundary_info=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

FV functional representing the inner product with an L2-Function on the boundary.

Parameters:

grid – Grid for which to assemble the functional.
function – The Function with which to take the inner product.
boundary_type – The type of domain boundary (e.g. ‘neumann’) on which to assemble the functional. If None the functional is assembled over the whole boundary.
boundary_info – If boundary_type is specified, the BoundaryInfo determining which boundary entity belongs to which physical boundary.
name – The name of the functional.

source[source]¶

sparse = False[source]¶

class pymor.discretizers.builtin.fv.DiffusionOperator(grid, boundary_info, diffusion_function=None, diffusion_constant=None, solver=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite Volume Diffusion Operator.

The operator is of the form

(Lu)(x) = c ∇ ⋅ [ d(x) ∇ u(x) ]

Parameters:

grid – The Grid over which to assemble the operator.
boundary_info – BoundaryInfo for the treatment of Dirichlet boundary conditions.
diffusion_function – The scalar-valued Function d(x). If None, constant one is assumed.
diffusion_constant – The constant c. If None, c is set to one.
solver – The Solver for the operator.
name – Name of the operator.

sparse = True[source]¶

class pymor.discretizers.builtin.fv.EngquistOsherFlux(flux, flux_derivative, gausspoints=5, intervals=1)[source]¶

Bases: NumericalConvectiveFlux

Engquist-Osher numerical flux.

If f is the analytical flux, and f' its derivative, the Engquist-Osher flux is given by:

F(U_in, U_out, normal, vol) = vol * [c^+(U_in, normal)  +  c^-(U_out, normal)]

                                   U_in
c^+(U_in, normal)  = f(0)⋅normal +  ∫   max(f'(s)⋅normal, 0) ds
                                   s=0

                                  U_out
c^-(U_out, normal) =                ∫   min(f'(s)⋅normal, 0) ds
                                   s=0

Parameters:

flux – Function defining the analytical flux f.
flux_derivative – Function defining the analytical flux derivative f'.
gausspoints – Number of Gauss quadrature points to be used for integration.
intervals – Number of subintervals to be used for integration.

Methods

`evaluate_stage1`
`evaluate_stage2`

evaluate_stage1(U, mu=None)[source]¶

evaluate_stage2(stage1_data, unit_outer_normals, volumes, mu=None)[source]¶

class pymor.discretizers.builtin.fv.InterpolationOperator(grid, function, order=0)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Vector-like L^2-projection interpolation Operator for finite volume spaces.

Parameters:

grid – The Grid on which to interpolate.
function – The Function to interpolate.
order – The quadrature order to compute the element-wise averages

linear = True[source]¶

source[source]¶

class pymor.discretizers.builtin.fv.L2Functional(grid, function=None, boundary_info=None, dirichlet_data=None, diffusion_function=None, diffusion_constant=None, neumann_data=None, order=1, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite volume functional representing the inner product with an L2-Function.

Additionally, boundary conditions can be enforced by providing dirichlet_data and neumann_data functions.

Parameters:

grid – Grid for which to assemble the functional.
function – The Function with which to take the inner product or None.
boundary_info – BoundaryInfo determining the Dirichlet and Neumann boundaries or None. If None, no boundary treatment is performed.
dirichlet_data – Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
diffusion_function – See DiffusionOperator. Has to be specified in case dirichlet_data is given.
diffusion_constant – See DiffusionOperator. Has to be specified in case dirichlet_data is given.
neumann_data – Function providing the Neumann boundary values. If None, constant-zero is assumed.
order – Order of the Gauss quadrature to use for numerical integration.
name – The name of the functional.

source[source]¶

sparse = False[source]¶

class pymor.discretizers.builtin.fv.L2Product(grid, solver=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Operator representing the L2-product between finite volume functions.

Parameters:

grid – The Grid for which to assemble the product.
solver – The Solver for the operator.
name – The name of the product.

sparse = True[source]¶

class pymor.discretizers.builtin.fv.LaxFriedrichsFlux(flux, lxf_lambda=1.0)[source]¶

Bases: NumericalConvectiveFlux

Lax-Friedrichs numerical flux.

If f is the analytical flux, the Lax-Friedrichs flux F is given by:

F(U_in, U_out, normal, vol) = vol * [normal⋅(f(U_in) + f(U_out))/2 + (U_in - U_out)/(2*λ)]

Parameters:

flux – Function defining the analytical flux f.
lxf_lambda – The stabilization parameter λ.

Methods

`evaluate_stage1`
`evaluate_stage2`

evaluate_stage1(U, mu=None)[source]¶

evaluate_stage2(stage1_data, unit_outer_normals, volumes, mu=None)[source]¶

class pymor.discretizers.builtin.fv.LinearAdvectionLaxFriedrichsOperator(grid, boundary_info, velocity_field, lxf_lambda=1.0, solver=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Linear advection finite Volume Operator using Lax-Friedrichs flux.

The operator is of the form

L(u, mu)(x) = ∇ ⋅ (v(x, mu)⋅u(x))

See LaxFriedrichsFlux for the definition of the Lax-Friedrichs flux.

Parameters:

grid – Grid over which to assemble the operator.
boundary_info – BoundaryInfo determining the Dirichlet and Neumann boundaries.
velocity_field – Function defining the velocity field v.
lxf_lambda – The stabilization parameter λ.
solver – The Solver for the operator.
name – The name of the operator.

class pymor.discretizers.builtin.fv.NonlinearAdvectionOperator(grid, boundary_info, numerical_flux, dirichlet_data=None, jacobian_delta=1e-07, solver=None, name=None)[source]¶

Bases: pymor.operators.interface.Operator

Nonlinear finite volume advection Operator.

The operator is of the form

L(u, mu)(x) = ∇ ⋅ f(u(x), mu)

Parameters:

grid – Grid for which to evaluate the operator.
boundary_info – BoundaryInfo determining the Dirichlet and Neumann boundaries.
numerical_flux – The NumericalConvectiveFlux to use.
dirichlet_data – Function providing the Dirichlet boundary values. If None, constant-zero boundary is assumed.
solver – The Solver for the operator.
name – The name of the operator.

Methods

`apply`	Apply the operator to a `VectorArray`.
`jacobian`	Return the operator's Jacobian as a new `Operator`.
`restricted`	Restrict the operator range to a given set of degrees of freedom.
`with_numerical_flux`

linear = False[source]¶

apply(U, mu=None)[source]¶

Apply the operator to a VectorArray.

Parameters:

U – VectorArray of vectors to which the operator is applied.
mu – The parameter values for which to evaluate the operator.

Returns:

|VectorArray| of the operator evaluations.

jacobian(U, mu=None)[source]¶

Return the operator’s Jacobian as a new Operator.

If the operator has a Solver, the Jacobian Operator will be equipped with the solver’s jacobian_solver.

Parameters:

U – Length 1 VectorArray containing the vector for which to compute the Jacobian.
mu – The parameter values for which to compute the Jacobian.

Returns:

Linear |Operator| representing the Jacobian.

restricted(dofs)[source]¶

Restrict the operator range to a given set of degrees of freedom.

This method returns a restricted version restricted_op of the operator along with an array source_dofs such that for any VectorArray U in self.source the following is true:

self.apply(U, mu).dofs(dofs)
    == restricted_op.apply(restricted_op.source.from_numpy(U.dofs(source_dofs))
                           mu).to_numpy()

Such an operator is mainly useful for empirical interpolation where the evaluation of the original operator only needs to be known for few selected degrees of freedom. If the operator has a small stencil, only few source_dofs will be needed to evaluate the restricted operator which can make its evaluation very fast compared to evaluating the original operator.

Parameters:

dofs – One-dimensional NumPy array of degrees of freedom in the operator range to which to restrict.

Returns:

restricted_op – The restricted operator as defined above. The operator will have NumpyVectorSpace (len(source_dofs)) as source and NumpyVectorSpace (len(dofs)) as range.
source_dofs – One-dimensional NumPy array of source degrees of freedom as defined above.

with_numerical_flux(**kwargs)[source]¶

class pymor.discretizers.builtin.fv.NonlinearReactionOperator(grid, reaction_function, reaction_function_derivative=None, name=None)[source]¶

Bases: pymor.operators.interface.Operator

Finite Volume nonlinear reaction Operator.

Parameters:

grid – The Grid for which to assemble the operator.
reaction_function – The reaction function.
reaction_function_derivative – The reaction function derivative.
name – The name of the operator.

Methods

`apply`	Apply the operator to a `VectorArray`.
`jacobian`	Return the operator's Jacobian as a new `Operator`.

apply(U, ind=None, mu=None)[source]¶

Apply the operator to a VectorArray.

Parameters:

U – VectorArray of vectors to which the operator is applied.
mu – The parameter values for which to evaluate the operator.

Returns:

|VectorArray| of the operator evaluations.

jacobian(U, mu=None)[source]¶

Return the operator’s Jacobian as a new Operator.

If the operator has a Solver, the Jacobian Operator will be equipped with the solver’s jacobian_solver.

Parameters:

U – Length 1 VectorArray containing the vector for which to compute the Jacobian.
mu – The parameter values for which to compute the Jacobian.

Returns:

Linear |Operator| representing the Jacobian.

class pymor.discretizers.builtin.fv.NumericalConvectiveFlux[source]¶

Bases: pymor.parameters.base.ParametricObject

Interface for numerical convective fluxes for finite volume schemes.

Numerical fluxes defined by this interfaces are functions of the form F(U_inner, U_outer, unit_outer_normal, edge_volume, mu).

The flux evaluation is vectorized and happens in two stages:

evaluate_stage1 receives a NumPy array U of all values which appear as U_inner or U_outer for all edges the flux shall be evaluated at and returns a tuple of NumPy arrays each of the same length as U.
evaluate_stage2 receives the reordered stage1_data for each edge as well as the unit outer normal and the volume of the edges.

stage1_data is given as follows: If R_l is l-th entry of the tuple returned by evaluate_stage1, the l-th entry D_l of of the stage1_data tuple has the shape (num_edges, 2) + R_l.shape[1:]. If for edge k the values U_inner and U_outer are the i-th and j-th value in the U array provided to evaluate_stage1, we have
```
D_l[k, 0] == R_l[i],    D_l[k, 1] == R_l[j].
```
evaluate_stage2 returns a NumPy array of the flux evaluations for each edge.

Methods

`evaluate_stage1`
`evaluate_stage2`

abstract evaluate_stage1(U, mu=None)[source]¶

abstract evaluate_stage2(stage1_data, unit_outer_normals, volumes, mu=None)[source]¶

class pymor.discretizers.builtin.fv.ReactionOperator(grid, reaction_coefficient, solver=None, name=None)[source]¶

Bases: pymor.operators.numpy.NumpyMatrixBasedOperator

Finite Volume reaction Operator.

The operator is of the form

L(u, mu)(x) = c(x, mu)⋅u(x)

Parameters:

grid – The Grid for which to assemble the operator.
reaction_coefficient – The function ‘c’
solver – The Solver for the operator.
name – The name of the operator.

sparse = True[source]¶

class pymor.discretizers.builtin.fv.SimplifiedEngquistOsherFlux(flux, flux_derivative)[source]¶

Bases: NumericalConvectiveFlux

Engquist-Osher numerical flux. Simplified Implementation for special case.

For the definition of the Engquist-Osher flux see EngquistOsherFlux. This class provides a faster and more accurate implementation for the special case that f(0) == 0 and the derivative of f only changes sign at 0.

Parameters:

flux – Function defining the analytical flux f.
flux_derivative – Function defining the analytical flux derivative f'.

Methods

`evaluate_stage1`
`evaluate_stage2`

evaluate_stage1(U, mu=None)[source]¶

evaluate_stage2(stage1_data, unit_outer_normals, volumes, mu=None)[source]¶

pymor.discretizers.builtin.fv.FVVectorSpace(grid)[source]¶

pymor.discretizers.builtin.fv.discretize_instationary_fv(analytical_problem, diameter=None, domain_discretizer=None, grid_type=None, num_flux='lax_friedrichs', lxf_lambda=1.0, eo_gausspoints=5, eo_intervals=1, grid=None, boundary_info=None, num_values=None, time_stepper=None, nt=None, preassemble=True, solver=None)[source]¶

FV Discretization of an InstationaryProblem with a StationaryProblem as stationary part.

Parameters:

analytical_problem – The InstationaryProblem to discretize.
diameter – If not None, diameter is passed to the domain_discretizer.
domain_discretizer – Discretizer to be used for discretizing the analytical domain. This has to be a function domain_discretizer(domain_description, diameter, ...). If further arguments should be passed to the discretizer, use functools.partial. If None, discretize_domain_default is used.
grid_type – If not None, this parameter is forwarded to domain_discretizer to specify the type of the generated Grid.
num_flux – The numerical flux to use in the finite volume formulation. Allowed values are 'lax_friedrichs', 'engquist_osher', 'simplified_engquist_osher' (see pymor.discretizers.builtin.fv).
lxf_lambda – The stabilization parameter for the Lax-Friedrichs numerical flux (ignored, if different flux is chosen).
eo_gausspoints – Number of Gauss points for the Engquist-Osher numerical flux (ignored, if different flux is chosen).
eo_intervals – Number of sub-intervals to use for integration when using Engquist-Osher numerical flux (ignored, if different flux is chosen).
grid – Instead of using a domain discretizer, the Grid can also be passed directly using this parameter.
boundary_info – A BoundaryInfo specifying the boundary types of the grid boundary entities. Must be provided if grid is specified.
num_values – The number of returned vectors of the solution trajectory. If None, each intermediate vector that is calculated is returned.
time_stepper – The time-stepper to be used by solve of InstationaryModel.
nt – If time_stepper is not specified, the number of time steps for implicit Euler time stepping.
preassemble – If True, preassemble all operators in the resulting Model.
solver – The Solver to be used.

Returns:

m – The Model that has been generated.
data – Dictionary with the following entries:

grid:

The generated Grid.

boundary_info:

The generated BoundaryInfo.

unassembled_m:

In case preassemble is True, the generated Model before preassembling operators.

pymor.discretizers.builtin.fv.discretize_stationary_fv(analytical_problem, diameter=None, domain_discretizer=None, grid_type=None, num_flux='lax_friedrichs', lxf_lambda=1.0, eo_gausspoints=5, eo_intervals=1, grid=None, boundary_info=None, preassemble=True, solver=None)[source]¶

Discretizes a StationaryProblem using the finite volume method.

Parameters:

analytical_problem – The StationaryProblem to discretize.
diameter – If not None, diameter is passed as an argument to the domain_discretizer.
domain_discretizer – Discretizer to be used for discretizing the analytical domain. This has to be a function domain_discretizer(domain_description, diameter, ...). If None, discretize_domain_default is used.
grid_type – If not None, this parameter is forwarded to domain_discretizer to specify the type of the generated Grid.
num_flux – The numerical flux to use in the finite volume formulation. Allowed values are 'lax_friedrichs', 'engquist_osher', 'simplified_engquist_osher' (see pymor.discretizers.builtin.fv).
lxf_lambda – The stabilization parameter for the Lax-Friedrichs numerical flux (ignored, if different flux is chosen).
eo_gausspoints – Number of Gauss points for the Engquist-Osher numerical flux (ignored, if different flux is chosen).
eo_intervals – Number of sub-intervals to use for integration when using Engquist-Osher numerical flux (ignored, if different flux is chosen).
grid – Instead of using a domain discretizer, the Grid can also be passed directly using this parameter.
boundary_info – A BoundaryInfo specifying the boundary types of the grid boundary entities. Must be provided if grid is specified.
preassemble – If True, preassemble all operators in the resulting Model.
solver – The Solver to be used.

Returns:

m – The Model that has been generated.
data – Dictionary with the following entries:

grid:

The generated Grid.

boundary_info:

The generated BoundaryInfo.

unassembled_m:

In case preassemble is True, the generated Model before preassembling operators.

pymor.discretizers.builtin.fv.nonlinear_advection_engquist_osher_operator(grid, boundary_info, flux, flux_derivative, gausspoints=5, intervals=1, dirichlet_data=None, jacobian_delta=None, solver=None, name=None)[source]¶: Instantiate a NonlinearAdvectionOperator using EngquistOsherFlux.

pymor.discretizers.builtin.fv.nonlinear_advection_lax_friedrichs_operator(grid, boundary_info, flux, lxf_lambda=1.0, dirichlet_data=None, jacobian_delta=None, solver=None, name=None)[source]¶: Instantiate a NonlinearAdvectionOperator using LaxFriedrichsFlux.

pymor.discretizers.builtin.fv.nonlinear_advection_simplified_engquist_osher_operator(grid, boundary_info, flux, flux_derivative, dirichlet_data=None, jacobian_delta=None, solver=None, name=None)[source]¶: Create a NonlinearAdvectionOperator using SimplifiedEngquistOsherFlux.

pymor.discretizers.builtin.fv¶

Module Contents¶

`pymor.discretizers.builtin.fv`¶