A second order uniform convergent method for a singularly perturbed parabolic system of reaction–diffusion type∗

In addition, we suppose that sufficient compatibility conditions among the data of the differential equation hold, in order that the exact solution ~u ∈ C4,3(Q̄), i.e, continuity up to fourth order in space and up to third order in time. This problem is a simple model of the classical linear double–diffusion model for saturated flow in fractured porous media (Barenblatt system) developed in [1]. Also this problem can be used to model diffusion process in bones (see [4]). It is well–known that the exact solution of these problems has a multiscale character, i.e., there are boundary layers. Therefore, it is necessary to dispose of efficient numerical methods (uniformly convergent methods) to approximate the solution independently of the values of the diffusion parameters ε1 and ε2. Recently some papers (see [7], [8], [9] [10] and [11]) study uniform convergent numerical methods to solve singularly perturbed elliptic linear systems on a special piecewise uniform Shishkin mesh, for different relations between the diffusion parameters: i) ε1 = ε, ε2 = 1; ii) ε1 = ε2 = ε; iii) ε1, ε2 arbitrary. Here, we are interested in increasing the uniform convergence order of the numerical method given in [6], which was used to solve a parabolic coupled system of type (1). With this aim we will combine the Crank-Nicolson method to discretize the time variable joint to the central finite differences discretization in space. Previously this method has been used in the framework of singularly perturbed problems; for instance, in [2] it was considered to solve a class of 1D parabolic problems of convection-diffusion type. We denote by Γ0 = {(x, 0) |x ∈ Ω}, Γ1 = {(x, t) |x = 0, 1, t ∈ [0, T ]}, Γ = Γ0 ∪ Γ1 and ~ε = (ε1, ε2) , with 0 < ε1 ≤ ε2 ≤ 1, the vectorial singular perturbation parameter. We write ∗This research will be partially supported by the project MEC/FEDER MTM 2004-01905 and the Diputación General de Aragón.