Freezing similarity solutions in multi-dimensional Burgers' Equation

TL;DR

This paper introduces a symmetry-based method to efficiently simulate and analyze similarity solutions in multi-dimensional Burgers' equations, enabling long-term simulations and observation of metastable behaviors.

Contribution

The authors derive a symmetry-based 'freezing system' for multi-dimensional Burgers' equations, facilitating long-time simulations and analysis of similarity solutions.

Findings

Effective long-term simulation of similarity solutions

Ability to observe metastable N-wave-like patterns

Derivation of symmetry-based 'freezing system'

Abstract

The topic of this paper are similarity solutions occurring in multi-dimensional Burgers' equation. We present a simple derivation of the symmetries appearing in a family of generalizations of Burgers' equation in $d$-space dimensions. These symmetries we use to derive an equivalent partial differential algebraic equation (freezing system) that allows us to do long time simulations and obtain good approximations of similarity solutions by direct forward simulation. The method also allows us without further effort to observe meta-stable behavior near N-wave-like patterns.