Adaptive multiresolution schemes with local time stepping for two-dimensional degenerate reaction-diffusion systems

TL;DR

This paper introduces an adaptive multiresolution scheme with local time stepping for two-dimensional reaction-diffusion systems, achieving efficient data compression and computational speedup while maintaining accuracy, especially in modeling sharp fronts and discontinuities.

Contribution

It develops a novel adaptive multiresolution method with local time stepping for degenerate reaction-diffusion systems, optimizing error control and computational efficiency.

Findings

Data compression and CPU time are significantly reduced.

Local time stepping doubles the acceleration without increasing error.

The scheme effectively captures steep gradients and sharp fronts.

Abstract

We present a fully adaptive multiresolution scheme for spatially two-dimensional, possibly degenerate reaction-diffusion systems, focusing on combustion models and models of pattern formation and chemotaxis in mathematical biology. Solutions of these equations in these applications exhibit steep gradients, and in the degenerate case, sharp fronts and discontinuities. The multiresolution scheme is based on finite volume discretizations with explicit time stepping. The multiresolution representation of the solution is stored in a graded tree. By a thresholding procedure, namely the elimination of leaves that are smaller than a threshold value, substantial data compression and CPU time reduction is attained. The threshold value is chosen optimally, in the sense that the total error of the adaptive scheme is of the same slope as that of the reference finite volume scheme. Since chemical reactions involve a large range of temporal scales, but are spatially well localized (especially in the combustion model), a locally varying adaptive time stepping strategy is applied. It turns out that local time stepping accelerates the adaptive multiresolution method by a factor of two, while the error remains controlled.