Arbitrary High-Order Unconditionally Stable Methods for Reaction-Diffusion Equations with inhomogeneous Boundary Condition via Deferred Correction
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Abstract
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> 2p+2 of accuracy at the stage <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>p</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>0</m:mn> <m:mo>,</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo>,</m:mo> <m:mi mathvariant="normal">…</m:mi> </m:mrow> </m:mrow> </m:math> p=0,1,2,\ldots 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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mo></m:mo> <m:mi>p</m:mi> </m:mrow> <m:mo>+</m:mo> <m:mn>2</m:mn> </m:mrow> </m:math> 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 a quasi-uniform family of meshes. Numerical tests with a bistable reaction-diffusion equation having a strong stiffness ratio, a Fisher equation, a linear reaction-diffusion equation addressing order reduction and two linear IBVPs in two dimensions are performed and demonstrate the unconditional convergence of the method. The orders 2, 4, 6, 8 and 10 of accuracy in time are achieved. Except for some linear problems, the accuracy of DC methods is better than that of BDF methods of same order.
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