implementations.problem_classes.AdvectionDiffusionEquation_1D_FFT module

class advectiondiffusion1d_imex(nvars=256, c=1.0, freq=-1, nu=0.02, L=1.0)

Bases: Problem

Example implementing the unforced one-dimensional advection diffusion equation

\[\frac{\partial u(x,t)}{\partial t} = - c \frac{\partial u(x,t)}{\partial x} + \nu \frac{\partial^2 u(x,t)}{\partial x^2}\]

with periodic boundary conditions in \([-\frac{L}{2}, \frac{L}{2}]\) in spectral space. The advection part \(- c \frac{\partial u(x,t)}{\partial x}\) is treated explicitly, whereas the diffusion part \(\nu \frac{\partial^2 u(x,t)}{\partial x^2}\) will be treated numerically in an implicit way. The exact solution is given by

\[u(x, t) = \sin(\omega (x - c t)) \exp(-t \nu \omega^2)\]

for \(\omega=2 \pi k\), where \(k\) denotes the wave number. Fast Fourier transform is used for the spatial discretization.

Parameters:

• nvars (int, optional) – Number of points in spatial discretization.

• c (float, optional) – Advection speed \(c\).

• freq (int, optional) – Wave number \(k\).

• nu (float, optional) – Diffusion coefficient \(\nu\).

• L (float, optional) – Denotes the period of the function to be approximated for the Fourier transform.

xvalues

Contains the grid points in space.

Type:

np.1darray

ddx

Spectral operator for gradient.

Type:

np.1darray

lap

Spectral operator for Laplacian.

Type:

np.1darray

dtype_f

alias of imex_mesh

dtype_u

alias of mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time at which the numerical solution is computed.

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Simple FFT solver for the diffusion part.

Parameters:

• rhs (dtype_f) – Right-hand side for the linear system.

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver (not used here so far).

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u

u_exact(t)

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

Parameters:

t (float) – Time of the exact solution.

Returns:

me – The exact solution.

Return type:

dtype_u

class advectiondiffusion1d_implicit(nvars=256, c=1.0, freq=-1, nu=0.02, L=1.0)

Bases: advectiondiffusion1d_imex

Example implementing the unforced one-dimensional advection diffusion equation

\[\frac{\partial u(x,t)}{\partial t} = - c \frac{\partial u(x,t)}{\partial x} + \nu \frac{\partial^2 u(x,t)}{\partial x^2}\]

with periodic boundary conditions in \([-\frac{L}{2}, \frac{L}{2}]\) in spectral space. This class implements the problem solving it with fully-implicit time-stepping. The exact solution is given by

\[u(x, t) = \sin(\omega (x - c t)) \exp(-t \nu \omega^2)\]

for \(\omega=2 \pi k\), where \(k\) denotes the wave number. Fast Fourier transform is used for the spatial discretization.

Note

This class has the same attributes as the class it inherits from.

dtype_f

alias of mesh

dtype_u

alias of mesh

eval_f(u, t)

Routine to evaluate the right-hand side of the problem.

Parameters:

• u (dtype_u) – Current values of the numerical solution.

• t (float) – Current time at which the numerical solution is computed.

Returns:

f – The right-hand side of the problem.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)

Simple FFT solver for the diffusion and advection part (both are linear!).

Parameters:

• rhs (dtype_f) – Right-hand side for the linear system.

• factor (float) – Abbrev. for the node-to-node stepsize (or any other factor required).

• u0 (dtype_u) – Initial guess for the iterative solver (not used here so far).

• t (float) – Current time (e.g. for time-dependent BCs).

Returns:

me – The solution as mesh.

Return type:

dtype_u