Burgers’ Problem#

Formulation of the Burgers’ problem.

class BurgersProblem(nu=0)[source]#

Bases: TimeDependentProblem, SpatialProblem

Implementation of the one-dimensional Burgers’ problem on the space-time domain \(\Omega\times T = [-1, 1] \times [0, 1]\).

The problem is governed by the Burgers’ equation

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

where \(u = u(x, t)\) is the solution field and \(\nu \geq 0\) is the viscosity coefficient. For \(\nu = 0\), the equation reduces to the inviscid Burgers’ equation.

Homogeneous Dirichlet boundary conditions are imposed at the spatial boundaries:

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

The initial condition is prescribed as

\[u(x, 0) = -\sin(\pi x), \qquad x \in [-1, 1].\]

See also

Original reference: Raissi M., Perdikaris P., Karniadakis G. E. (2017). Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations. DOI: 10.48550.

Example:
>>> problem = BurgersProblem()

Initialization of the BurgersProblem class.

Parameters:

nu (float | int) – The viscosity coefficient.

Raises: