An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers' equation

TL;DR

This paper introduces an IMEX-RK numerical scheme for efficiently capturing similarity solutions in multidimensional Burgers' equation, demonstrating second order convergence and applicability to hyperbolic-parabolic PDEs.

Contribution

The paper presents a new IMEX-RK scheme for the freezing method, with proven second order convergence and effectiveness across various viscosity regimes.

Findings

Second order convergence for the time discretization.

Effective for all viscosity sizes in Burgers' equation.

Suitable for general hyperbolic-parabolic PDEs.

Abstract

In this paper we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations. The scheme is based on an IMEX-Runge-Kutta approach for a method of lines (semi-)discretization of the freezing partial differential algebraic equation (PDAE). We prove second order convergence for the time discretization at smooth solutions in the ODE-sense and we present numerical experiments that show second order convergence for the full discretization of the PDAE. As an example serves the multi-dimensional Burgers' equation. By considering very different sizes of viscosity, Burgers' equation can be considered as a prototypical example of general coupled hyperbolic-parabolic PDEs. Numerical experiments show that our method works perfectly well for all sizes of viscosity, suggesting that the scheme is indeed suitable for capturing similarity solutions in general hyperbolic-parabolic PDEs by direct forward simulation with the freezing method.