Arbitrary high-order unconditionally stable methods for reaction-diffusion equations with inhomogeneous boundary condition via Deferred Correction

TL;DR

This paper develops high-order, unconditionally stable numerical methods for reaction-diffusion equations with inhomogeneous boundary conditions, using deferred correction and finite element discretization, achieving accuracy up to order 10.

Contribution

It introduces a novel full discretization scheme that combines deferred correction with finite element methods for reaction-diffusion IBVPs, ensuring high-order accuracy and unconditional stability.

Findings

Achieves time accuracy orders 2,4,6,8,10.

Proves unconditional stability of the scheme.

Demonstrates effectiveness through numerical tests.

Abstract

In this paper we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. To avoid possible order reduction, the IBVP is first transformed into an IBVP with homogeneous boundary conditions (IBVPHBC) via a lifting of inhomogeneous Dirichlet, Neumann or mixed Dirichlet-Neumann boundary conditions. The IBVPHBC is discretized in time via the deferred correction method for the implicit midpoint rule and leads to a time-stepping scheme of order $2p+2$ of accuracy at the stage $p=0,1,2,\cdots $ of the correction. Each semi-discretized scheme results in a nonlinear elliptic equation for which the existence of a solution is proven using the Schaefer fixed point theorem. The elliptic equation corresponding to the stage $p$ of the correction is discretized by the Galerkin finite element method and gives a full discretization of the IBVPHBC. This fully discretized scheme is unconditionally stable with order $2p+2$ of accuracy in time. The order of accuracy in space is equal to the degree of the finite element used when the family of meshes considered is shape-regular while an increment of one order is proven for quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio and a linear reaction-diffusion equation addressing order reduction are performed and demonstrate the unconditional convergence of the method. The orders 2,4,6,8 and 10 of accuracy in time are achieved.