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 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.
Highlights
The preservation of the structure of nonlinear diffusion equations on the discrete level is of paramount importance in applications
We analyze a toy problem, the Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation with no-flux boundary conditions, to devise an implicit Euler discontinuous Galerkin scheme which preserves the positivity of the solution, the entropy structure, and the exponential equilibration on the discrete level
The Fisher–KPP equation admits traveling-wave solutions u(x, t) = φ(x − ct), which switch between the unstable steady state u∗ = 0 and the stable steady state u∗ = 1
Summary
The preservation of the structure of nonlinear diffusion equations on the discrete level is of paramount importance in applications. We analyze a toy problem, the Fisher–Kolmogorov–Petrovsky–Piscounov (Fisher–KPP) equation with no-flux boundary conditions, to devise an implicit Euler discontinuous Galerkin scheme which preserves the positivity of the solution, the entropy structure, and the exponential equilibration on the discrete level. This condition implies a positive lower bound for the total mass eλkh d x, which is needed to guarantee that the discrete solution converges to the stable steady state u∗ = 1 and not to the steady state u∗ = 0. Compared to conforming finite-element methods, DG methods allow for a more flexible mesh design and polynomial degree distribution, are easier to parallelize, allow to better cope with data discontinuities (e.g. of the material coefficients or initial conditions), and are able to locally reproduce conservation properties They directly produce block-diagonal (or even diagonal) mass matrices, which is an advantage in time-dependent problems.
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