ADER finite volume schemes for nonlinear reaction–diffusion equations

Abstract

We construct finite volume schemes of arbitrary order of accuracy in space and time for solving nonlinear reaction–diffusion partial differential equations. The numerical schemes, written in conservative form, result from extending the Godunov and the ADER frameworks, both originally developed for approximating solutions to hyperbolic equations. The task is to define numerical fluxes and numerical sources. In the ADER approach, numerical fluxes are computed from solutions to the Derivative Riemann Problem (DRP) (or generalized Riemann problem, or high-order Riemann problem), the Cauchy problem in which the initial conditions either side of the interface are smooth functions, polynomials of arbitrary degree, for example. We propose, and systematically asses, a general DRP solver for nonlinear reaction–diffusion equations and construct corresponding finite volume schemes of arbitrary order of accuracy. Schemes of 1st to 10-th order of accuracy in space and time are implemented and systematically assessed, with particular attention paid to their convergence rates. Numerical examples are also given.