Analytical-Numerical Approach to Describing Time-Periodic Motion of Fronts in Singularly Perturbed Reaction–Advection–Diffusion Models

Abstract

The paper presents an analytical-numerical approach to the study of moving fronts in singularly perturbed reaction–diffusion–advection models. A method for generating a dynamically adapted grid for the efficient numerical solution of problems of this class is proposed. The method is based on a priori information about the motion and properties of the front, obtained by rigorous asymptotic analysis of a singularly perturbed parabolic problem. In particular, the essential parameters taken into account when constructing the grid are estimates of the position of the transition layer, as well as its width and structure. The proposed analytical-numerical approach can significantly save computer resources, reduce the computation time, and increase the stability of the computational process in comparison with the classical approaches. An example demonstrating the main ideas and methods of application of the proposed approach is considered.