Freezing similarity solutions in the multidimensional Burgers equation

The topic of this paper is similarity solutions occurring in the multidimensional Burgers equation and their numerical approximation. We present a simple derivation of symmetries appearing in a family of generalizations of Burgers' equation in d-space dimensions. We use these symmetries to obtain an equivalent partial differential algebraic equation (freezing system) that allows us to do direct long-time simulations on a fixed computational domain. As concrete examples, we are able with this method to numerically find similarity solutions for the 2D Burgers equation and observe a metastable behavior of 2D N-wave-like patterns.