Robust Numerical Method for a System of Singularly Perturbed Parabolic Reaction‐diffusion Equations on a Rectangle

Abstract

Abstract. A Dirichlet problem is considered for a system of two singularly perturbed parabolic reaction-diffusion equations on a rectangle. The parabolic boundary layer appears in the solution of the problem as the perturbation parameter ε tends to zero. On the basis of the decomposition solution technique, estimates for the solution and derivatives are obtained. Using the condensing mesh technique and the classical finite difference approximations of the boundary value problem under consideration, a difference scheme is constructed that converges ε-uniformly at the rate0´, where N = mins Ns, s = 1, 2, Ns + 1 and N0 + 1 are the numbers of mesh points on the axis xs and on the axis t, respectively.

A coupled system of singularly perturbed parabolic reaction-diffusion equations

“…Shishkin [21] and Shishkina and Shishkin [22] also obtained second order convergence (ignoring logarithmic factors) for both the steady-state [21] and the time dependent problem [22] in two space dimensions, with m = 2, ε 1 = ε 2 and assuming that the coupling matrix A is strictly diagonally dominant with positive diagonal elements, which is a weaker assumption on the coupling matrix A than the assumptions in (1.2b)-(1.3). The time dependent problem (1.1) with two equations has been analyzed in [4,20].…”

A robust finite difference scheme for strongly coupled systems of singularly perturbed convection‐diffusion equations

“…Then from (2.23), we obtain the IAS scheme for the 1D coupled system (2.1) at the grid point x i , 25) or equivalently,…”

“…However, its convergence is not uniform in the perturbation parameter ε in the discrete maximum norm [3]. Although one can probably design some certain layer-adapted meshes [5,17,[23][24][25] to achieve the uniform convergence property, it seems not an easy task because until now not much is known in the literature about the structure of layers of the strongly coupled system (1.1), see [2,26] and many references cited therein.…”

Conservative Numerical Method for a System of Semilinear Singularly Perturbed Parabolic Reaction‐diffusion Equations

“…A boundary value problem on a rectangle for a system of linear parabolic reaction-diffusion equations have been considered in [15,17] and for a system of linear elliptic reaction-diffusion equations with two perturbation parameters have been considered in [13,16].…”

“…A priori estimates for the problem solutions and their regular and singular components that are needed for the construction and study of difference schemes are exposed in Section 4. To derive a priori estimates and justify convergence of special finite difference schemes, a technique is applied that had been developed for a system of linear singularly perturbed elliptic [13,16] and parabolic [15,17] equations on a rectangle. A nonlinear conservative difference scheme on the rectangular grid with an arbitrary distribution of nodes (in particular, on uniform grid) is constructed in Section 5.…”