implementations.problem_classes.AcousticAdvection_1D_FD_imex module

class acoustic_1d_imex(nvars=None, cs=0.5, cadv=0.1, order_adv=5, waveno=5)[source]

Bases: ptype

This class implements the one-dimensional acoustics advection equation on a periodic domain \([0, 1]\) fully investigated in [1]. The equations are given by

\[\frac{\partial u}{\partial t} + c_s \frac{\partial p}{\partial x} + U \frac{\partial u}{\partial x} = 0,\]
\[\frac{\partial p}{\partial t} + c_s \frac{\partial u}{\partial x} + U \frac{\partial p}{\partial x} = 0.\]

For initial data \(u(x, 0) \equiv 0\) and \(p(x, 0) = p_0 (x)\) the analytical solution is

\[u(x, t) = \frac{1}{2} p_0 (x - (U + c_s) t) - \frac{1}{2} p_0 (x - (U - c_s) t),\]
\[p(x, t) = \frac{1}{2} p_0 (x - (U + c_s) t) + \frac{1}{2} p_0 (x - (U - c_s) t).\]

The problem is implemented in the way that is used for IMEX time-stepping.

Parameters:

  • nvars (int, optional) – Number of degrees of freedom.

  • cs (float, optional) – Sound velocity \(c_s\).

  • cadv (float, optional) – Advection speed \(U\).

  • order_adv (int, optional) – Order of which the advective derivative is discretized.

  • waveno (int, optional) – The wave number.

mesh

1d mesh.

Type:

np.1darray

dx

Mesh size.

Type:

float

Dx

Matrix for the advection operator.

Type:

scipy.csc_matrix

Id

Sparse identity matrix.

Type:

scipy.csc_matrix

A

Matrix for the wave operator.

Type:

scipy.csc_matrix

References

dtype_f

alias of imex_mesh

dtype_u

alias of mesh

eval_f(u, t)[source]

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

Parameters:

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

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

Returns:

f – The right-hand side divided into two parts.

Return type:

dtype_f

solve_system(rhs, factor, u0, t)[source]

Simple linear solver for \((I-factor\cdot A)\vec{u}=\vec{rhs}\).

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)[source]

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