AdvectionProblem

Formulation of the advection problem.

class AdvectionProblem(c=1.0)

Bases: SpatialProblem, TimeDependentProblem

Implementation of the one-dimensional advection problem on the space-time domain \(\Omega\times T = [0, 2\pi] \times [0, 1]\).

The problem is governed by the linear advection equation

\[\frac{\partial u}{\partial t} + c \frac{\partial u}{\partial x} = 0,\]

where \(u = u(x, t)\) is the solution field and \(c\) is the advection velocity.

Periodic boundary conditions are imposed at the spatial boundaries:

\[u(0, t) = u(2\pi, t), \qquad t \in [0, 1].\]

The initial condition is prescribed as

\[u(x, 0) = \sin(x), \qquad x \in [0, 2\pi].\]

The analytical solution is given by

\[u(x, t) = \sin(x - ct).\]

See also

Original reference: Wang, Sifan, et al. An expert’s guide to training physics-informed neural networks. arXiv preprint arXiv:2308.08468 (2023). DOI: arXiv:2308.08468.

Example:

>>> problem = AdvectionProblem()

Initialization of the AdvectionProblem.

Parameters:

c (float | int) – The advection velocity parameter. Default is 1.0.

solution(pts)

Implementation of the analytical solution of the advection problem.

Parameters:

pts (LabelTensor) – Points where the solution is evaluated.

Returns:

The analytical solution of the advection problem.

Return type:

LabelTensor