AN ALMOST 4TH ORDER UNIFORMLY CONVERGENT DIFFERENCE SCHEME FOR A SEMILINEAR SINGULARLY PERTURBED REACTION-DIFFUSION PROBLEM

This paper is concerned with a high order convergent discretization for the semilinear reaction-diffusion problem: -epsilon(2)u''+b(x,u)=O, for x is an element of (0, 1), subject to u(0)=u(1)=0, where epsilon is an element of (0, 1]. We assume that b(u)(x, u) > b(0)(2) > 0 On [0, 1] x R(1), which guarantees uniqueness of a solution to the problem. Asymptotic properties of this solution are discussed. We consider a polynomial-based three-point difference scheme on a simple piecewise equidistant mesh of Shishkin type. Existence and local uniqueness of a solution to the scheme are analysed. We prove that the scheme is almost fourth order accurate in the discrete maximum norm, uniformly in the perturbation parameter epsilon. We present numerical results in support of this result.