Optimized high-order IMEX-RK schemes for degenerate diffusion-reaction problems with application to travelling waves phenomena

Abstract

We study a class of IMplicit-EXplicit Runge--Kutta (IMEX-RK) schemes for the numerical approximation of reaction and diffusion-reaction problems arising in a variety of biological and physical applications. Such models may admit travelling wave solutions, with the Fisher--Kolmogorov equation representing a prototypical example. Motivated by this feature, the proposed time integration schemes are designed to accurately capture sharp propagating fronts. We also investigate a less standard use of IMEX-RK methods that circumvents a splitting of reaction terms into linear and nonlinear components, while still requiring the solution of linear systems at each stage. This semi-implicit formulation, referred to as SI-IMEX-RK, enables a targeted treatment of stiffness by isolating its relevant contributions. The time discretization is coupled with a high-order polygonal discontinuous Galerkin method for space discretization, resulting in a flexible and robust framework for the treatment of multiscale dynamics in complex geometries. A comprehensive validation strategy is presented to assess the accuracy and stability properties of the proposed schemes across a hierarchy of increasingly challenging test problems.