A structure-preserving discontinuous Galerkin scheme for the Fischer-KPP equation

TL;DR

This paper introduces a structure-preserving discontinuous Galerkin scheme for the Fisher-KPP equation, ensuring positivity, entropy decay, and convergence, with numerical validation in one dimension.

Contribution

It develops an implicit Euler DG scheme that preserves positivity and entropy decay for the Fisher-KPP equation, with rigorous convergence analysis.

Findings

The scheme maintains positivity of the discrete solution.

Discrete entropy decays over time, leading to stability.

Numerical experiments confirm theoretical convergence and properties.

Abstract

An implicit Euler discontinuous Galerkin scheme for the Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation for population densities with no-flux boundary conditions is suggested and analyzed. Using an exponential variable transformation, the numerical scheme automatically preserves the positivity of the discrete solution. A discrete entropy inequality is derived, and the exponential time decay of the discrete density to the stable steady state in the L1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L2 norm to the unique strong solution to the time-discrete Fisher-KPP equation as the mesh size tends to zero. Numerical experiments in one space dimension illustrate the theoretical results.