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

摘要

In this article, we introduce a new, simple and efficient numerical scheme for the implementation of the freezing method for capturing similarity solutions in partial differential equations (PDEs). The scheme is based on an implicit-explicit (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 ordinary differential equation sense, and we present numerical experiments that show second-order convergence for the full discretization of the PDAE. The multidimensional Burgers's equations serves as an example. By considering very different values of viscosity, Burgers's equation can be considered a prototypical example of general coupled hyperbolic-parabolic PDEs. Numerical experiments show that our method works perfectly well for all values 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.