An almost third order finite difference scheme for singularly perturbed reaction–diffusion systems

AbstractThis paper addresses the numerical approximation of solutions to coupled systems of singularly perturbed reaction–diffusion problems. In particular a hybrid finite difference scheme of HODIE type is constructed on a piecewise uniform Shishkin mesh. It is proved that the numerical scheme satisfies a discrete maximum principle and also that it is third order (except for a logarithmic factor) uniformly convergent, even for the case in which the diffusion parameter associated with each equation of the system has a different order of magnitude. Numerical examples supporting the theory are given

High order schemes for reaction-diffusion singularly perturbed systems

In this paper we are interested in solving e¿ciently a singularly per-turbed linear system of di¿erential equations of reaction-di¿usion type. Firstly, anon–monotone ¿nite di¿erence scheme of HODIE type is constructed on a Shishkinmesh. The previous method is modi¿ed at the transition points such that an inversemonotone scheme is obtained. We prove that if the di¿usion parameters are equal itis a third order uniformly convergent method. If the di¿usion parameters are di¿er-ent some numerical evidence is presented to suggest that an uniformly convergentscheme of order greater than two is obtained. Nevertheless, the uniform errors arebigger and the orders of uniform convergence are less than in the case correspondingto equal di¿usion parameters

A uniformly convergent scheme for a system of reaction–diffusion equations

AbstractIn this work a system of two parabolic singularly perturbed equations of reaction–diffusion type is considered. The asymptotic behaviour of the solution and its partial derivatives is given. A decomposition of the solution in its regular and singular parts has been used for the asymptotic analysis of the spatial derivatives. To approximate the solution we consider the implicit Euler method for time stepping and the central difference scheme for spatial discretization on a special piecewise uniform Shishkin mesh. We prove that this scheme is uniformly convergent, with respect to the diffusion parameters, having first-order convergence in time and almost second-order convergence in space, in the discrete maximum norm. Numerical experiments illustrate the order of convergence proved theoretically

On a decoupled algorithm for poroelasticity and its resolution by multigrid

In this work, we present a robust and efficient multigrid solver for a reformulated version of the system of poroelasticity equations. The reformulation enables us to treat the system in a decoupled fashion. It is sufficient to choose a highly efficient multigrid method for a scalar Poisson type equation for the overall solution of this poroelasticity system. With standard geometric transfer operators, a direct coarse grid discretization and a point-wise red-black Gauss-Seidel smoother, an efficient multigrid method is developed for all relevant choices of the problem parameters. According to classical multigrid theory, we observe multigrid convergence factors that are less than 0.1 for a variety of poroelastic problems. Very satisfactory solution times are produced, that are about 10 times faster than the iterative solution of the coupled system, discretized on a staggered grid

On a decoupled algorithm for poroelasticity and its resolution by multigrid

In this work, we present a robust and efficient multigrid solver for a reformulated version of the system of poroelasticity equations. The reformulation enables us to treat the system in a decoupled fashion. It is sufficient to choose a highly efficient multigrid method for a scalar Poisson type equation for the overall solution of this poroelasticity system. With standard geometric transfer operators, a direct coarse grid discretization and a point-wise red-black Gauss-Seidel smoother, an efficient multigrid method is developed for all relevant choices of the problem parameters. According to classical multigrid theory, we observe multigrid convergence factors that are less than 0.1 for a variety of poroelastic problems. Very satisfactory solution times are produced, that are about 10 times faster than the iterative solution of the coupled system, discretized on a staggered grid.Electrical Engineering, Mathematics and Computer Scienc