Arbitrary High Order A-Stable Time-Stepping Methods Via Deferred Correction: Application to Reaction-Diffusion Equations

Abstract

In this thesis we investigate time-stepping methods having both high order of accuracy and a good stability, for the numerical analysis of reaction-diffusion equations. The approach consists in a generalization and improvement of a time-stepping method introduced by Gustafson & Kress (2002). The time-stepping schemes from Gustafsson and Kress are built via a deferred correction (DC) strategy consisting in a successive correction (perturbation) of the trapezoidal rule, leading to a scheme of order 2j+2 of accuracy at the stage j=0,1,2, ... of the correction. However, this method addresses only linear initial-value problems (IVP) satisfying a monotonicity condition while it has an issue for the starting procedure, and it does not take advantage of an exhaustive convergence analysis for its applicability to stiff problems. Our approach is executed into three essential steps, leading to three submitted articles. First, we introduce general formulae to derive suitable arbitrary high order finite difference approximations of analytic functions. New forms of finite difference formulae suited to various approaches of DC time-stepping schemes and the computation of their starting values, complying with the high order requirement, are constructed. Second, we introduce a general idea for the construction of different DC schemes, and we present our time-stepping method. The time-stepping method consists in a sequence {DC2j} of self-starting schemes built recursively from the implicit midpoint rule via the DC strategy. A complete analysis of convergence of the method, in the case of general ordinary differential equations (ODE), is given using a deferred correction condition which guarantees an improvement by two of the order of accuracy while each scheme DC2j is corrected to get the scheme DC(2j+2). We prove that each DC2j is A-stable. Finally, we apply our DC method to an initial boundary value problem (IBVP) related to a large class of reaction-diffusion system. The IBVP is first discretized in the time variable via the DC method, and it follows a discretization in space by the Galerkin finite element method. We prove that the resulting schemes are unconditionally and strongly stable with order 2j+2 of accuracy in time (at the stage j of the correction). The order of accuracy in space is at least equal to the degree of the finite element used. All the theories, for ODEs and IBVP, are supported by numerical tests on various standard problems with the schemes DC2, ..., DC10. The numerical experiments comply with the theory and show that the theoretical orders of accuracy are always achieved together with a satisfactory stability.