An IMEX-RK scheme for capturing similarity solutions in the multidimensional Burgers' equation
This work provides a practical numerical tool for researchers studying similarity solutions in hyperbolic-parabolic PDEs, but the contribution is incremental as it applies known IMEX-RK techniques to a specific method.
The paper introduces a new IMEX-Runge-Kutta scheme for implementing the freezing method to capture similarity solutions in PDEs, achieving second-order convergence and working well across all viscosity sizes for the multidimensional Burgers' equation.
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.