Arbitrary high-order unconditionally stable methods for reaction-diffusion equations via Deferred Correction: Case of the implicit midpoint rule
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In this paper we analyse full discretizations of an initial boundary value problem (IBVP) related to reaction-diffusion equations. The IBVP is first 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,... 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 IBVP. This fully discretized scheme is unconditionlly 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 shape-regular and quasi-uniform family of meshes. A numerical test with a bistable reaction-diffusion equation having a strong stiffness ratio is performed and shows that the orders 2,4,6,8 and 10 of accuracy in time are achieved with a very strong stability.
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