implementations.problem_classes.AdvectionEquation_ND_FD module

class advectionNd(nvars=512, c=1.0, freq=2, stencil_type='center', order=2, lintol=1e-12, liniter=10000, solver_type='direct', bc='periodic', sigma=0.06)[source]

Bases: GenericNDimFinDiff

Example implementing the unforced ND advection equation with periodic or Dirichlet boundary conditions in \([0,1]^N\)

\[\frac{\partial u}{\partial t} = -c \frac{\partial u}{\partial x},\]

and initial solution of the form

\[u({\bf x},0) = \prod_{i=1}^N \sin(f\pi x_i),\]

with \(x_i\) the coordinate in \(i^{th}\) dimension. Discretization uses central finite differences.

Parameters:

  • nvars (int of tuple, optional) – Spatial resolution (same in all dimensions). Using a tuple allows to consider several dimensions, e.g nvars=(16,16) for a 2D problem.

  • c (float, optional) – Advection speed (same in all dimensions).

  • freq (int of tuple, optional) – Spatial frequency \(f\) of the initial conditions, can be tuple.

  • stencil_type (str, optional) – Type of the finite difference stencil.

  • order (int, optional) – Order of the finite difference discretization.

  • lintol (float, optional) – Tolerance for spatial solver (GMRES).

  • liniter (int, optional) – Max. iterations number for GMRES.

  • solver_type (str, optional) – Solve the linear system directly or using GMRES or CG

  • bc (str, optional) – Boundary conditions, either 'periodic' or 'dirichlet'.

  • sigma (float, optional) –

    If freq=-1 and ndim=1, uses a Gaussian initial solution of the form

    \[u(x,0) = e^{ \frac{\displaystyle 1}{\displaystyle 2} \left( \frac{\displaystyle x-1/2}{\displaystyle \sigma} \right)^2 }\]

A

FD discretization matrix of the ND grad operator.

Type:

sparse matrix (CSC)

Id

Identity matrix of the same dimension as A.

Type:

sparse matrix (CSC)

Note

Args can be set as values or as tuples, which will increase the dimension. Do, however, take care that all spatial parameters have the same dimension.

u_exact(t, **kwargs)[source]

Routine to compute the exact solution at time \(t\).

Parameters:

  • t (float) – Time of the exact solution.

  • **kwargs (dict) – Additional arguments (that won’t be used).

Returns:

sol – The exact solution.

Return type:

dtype_u