High order relaxed schemes for nonlinear reaction diffusion problems in nonconservative form

TL;DR

This paper develops high-order relaxed numerical schemes combining ENO/WENO spatial discretization with IMEX time integration to accurately solve nonlinear reaction-diffusion equations in nonconservative form, including degenerate cases.

Contribution

It introduces a nonstandard relaxation scheme for nonconservative diffusion equations and couples it with high-order methods, demonstrating improved accuracy and efficiency.

Findings

Effective high-order schemes for nonlinear reaction-diffusion equations.

Accurate handling of degenerate cases and nonconservative forms.

Successful application to ecological PDE models.

Abstract

Different relaxation approximations to partial differential equations, including conservation laws, Hamilton-Jacobi equations, convection-diffusion problems, gas dynamics problems, have been recently proposed. The present paper focuses onto diffusive relaxed schemes for the numerical approximation of nonlinear reaction diffusion equations. We choose here a nonstandard relaxation scheme that allow the treatment of diffusion equations in their nonconservative form. A comparison with the traditional approach in the case of conservative equations is also included. High order methods are obtained by coupling ENO and WENO schemes for space discretization with IMEX schemes for time integration, where the implicit part can be explicitly solved at a linear cost. To illustrate the high accuracy and good properties of the proposed numerical schemes, also in the degenerate case, we consider various examples in one and two dimensions: the Fisher-Kolmogoroff equation, the porous-Fisher equation and the porous medium equation with strong absorption. Moreover we show a test on a system of PDEs that describe an ecological model for the dispersal and settling of animal populations.