Burgers Equation 1D

anypinn create my-project --template burgers-1d

1D nonlinear PDE with shock formation. Recovers viscosity \(\nu\) with adaptive collocation.

Background

The viscous Burgers equation combines nonlinear advection (\(u\,\partial u / \partial x\)) with diffusion (\(\nu\,\partial^2 u / \partial x^2\)). It serves as a one-dimensional prototype for the Navier-Stokes equations and is a standard benchmark for methods that must handle shock formation. As the viscosity \(\nu \to 0\) the solution develops steep gradients that evolve into shock-like fronts, concentrating the PDE residual in narrow regions. This motivates the use of adaptive collocation, which places more training points where the residual is largest.

Governing Equations

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

where:

• \(u(x, t)\): velocity field
• \(\nu\): kinematic viscosity (to recover)

Default Configuration

The generated template uses the following values.

Parameters to recover:

Symbol	Code constant	True value
\(\nu\)	TRUE_NU	\(0.01 / \pi \approx 0.00318\)

Initial condition: \(u(x, 0) = -\sin(\pi x)\)

Boundary conditions: \(u(-1, t) = u(1, t) = 0\) (homogeneous Dirichlet).

Domain: \(x \in [-1, 1], \quad t \in [0, 1]\)

Features Demonstrated

• Adaptive collocation via AdaptiveCollocationCallback
• Shock-forming nonlinear PDE
• Scalar parameter recovery (viscosity)

Results