Approximation of a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle

Abstract

The Dirichlet problem for a system of singularly perturbed reaction-diffusion parabolic equations in a rectangle is considered. The higher order derivatives of the equations are multiplied by a perturbation parameter ɛ2, where ɛ takes arbitrary values in the interval (0, 1]. When ɛ vanishes, the system of parabolic equations degenerates into a system of ordinary differential equations with respect to t. When ɛ tends to zero, a parabolic boundary layer with a characteristic width ɛ appears in a neighborhood of the boundary. Using the condensing grid technique and the classical finite difference approximations of the boundary value problem, a special difference scheme is constructed that converges ɛ-uniformly at a rate of O(N −2ln2 N + N 0 −1 , where \( N = \mathop {\min }\limits_s N_s \), N s + 1 and N 0 + 1 are the numbers of mesh points on the axes x s and t, respectively.