A structure-preserving discontinuous Galerkin scheme for the Fisher–KPP equation

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 L-1 norm is proved if the initial entropy is smaller than the measure of the domain. The discrete solution is proved to converge in the L-2 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.