4.3.10 pde.pdes.wave module

A simple wave equation.

class WavePDE(speed=1, *, bc=None)[source]

Bases: PDEBase

A simple wave equation.

The mathematical definition, \(\partial_t^2 u = c^2 \nabla^2 u\), is implemented as two first-order equations,

\[\begin{split}\partial_t u &= v \\ \partial_t v &= c^2 \nabla^2 u\end{split}\]

where \(c\) sets the wave speed and \(v\) is an auxiallary field. Note that the class expects an initial condition specifying both fields, which can be created using the WavePDE.get_initial_condition() method. The result will also return two fields.

Parameters:

  • speed (float) – The speed \(c\) of the wave

  • bc (BoundariesData | None) – The boundary conditions applied to the field \(u\). Boundary conditions are generally given as a dictionary with one condition for each axis side. For periodic axes, only periodic boundary conditions are allowed (indicated by ‘periodic’ and ‘anti-periodic’). For non-periodic axes, different boundary conditions can be specified for the lower and upper end (using specific identifiers, like x- and y+). For instance, Dirichlet conditions enforcing a value NUM (specified by {‘value’: NUM}) and Neumann conditions enforcing the value DERIV for the derivative in the normal direction (specified by {‘derivative’: DERIV}) are supported. Note that the special value ‘auto_periodic_neumann’ imposes periodic boundary conditions for periodic axis and a vanishing derivative otherwise. More information can be found in the boundaries documentation.

default_bc = 'auto_periodic_neumann'

Default boundary condition used when no specific conditions are chosen.

evolution_rate(state, t=0)[source]

Evaluate the right hand side of the PDE.

Parameters:

  • state (FieldCollection) – The fields \(u\) and \(v\)

  • t (float) – The current time point

Returns:

Fields describing the evolution rates of the PDE

Return type:

FieldCollection

explicit_time_dependence: bool | None = False

Flag indicating whether the right hand side of the PDE has an explicit time dependence.

Type:

bool

property expressions: dict[str, str]

the expressions of the right hand side of this PDE

Type:

dict

get_initial_condition(u, v=None)[source]

Create a suitable initial condition.

Parameters:

  • u (ScalarField) – The initial density on the grid

  • v (ScalarField, optional) – The initial rate of change. This is assumed to be zero if the value is omitted.

Returns:

The combined fields u and v, suitable for the simulation

Return type:

FieldCollection