Stabilised approximation of interior-layer solutions of a singularly perturbed semilinear reaction–diffusion problem

Abstract

Abstract.A semilinear reaction-diffusion two-point boundary value problem, whose secondorder derivative is multiplied by a small positive parameter ε 2 , is considered. It can have multiple solutions. The numerical computation of solutions having interior transition layers is analysed. It is demonstrated that the accurate computation of such solutions is exceptionally difficult.To address this difficulty, we propose an artificial-diffusion stabilization. For both standard and stabilised finite difference methods on suitable Shishkin meshes, we prove existence and investigate the accuracy of computed solutions by constructing discrete sub-and super-solutions. Convergence results are deduced that depend on the relative sizes of ε and N , where N is the number of mesh intervals. Numerical experiments are given in support of these theoretical results. Practical issues in using Newton's method to compute a discrete solution are discussed.

The Capriciousness of Numerical Methods for Singular Perturbations

“…This observation is the basis for a postprocessing procedure [11] that generates an accurate numerical solution. In our numerical examples,C ≈ 2.2 and we can construct an approximation to the exact solution by shifting the numerical solution 2.2ε to the right; see Figure 3.6 for a close-up of the numerical solutions around x * N = x * = 0.5.…”

“…The important point x * is characterized by J(x * ) = 0. A standard example from the literature (see, e.g., [11]) is the boundary value problem…”

“…Now we solve (3.7) numerically using the central difference scheme on a Shishkin mesh for the shock layer region. For a given value x * N ∈ (0, 1) a fine equidistant mesh using N/2 intervals is defined on [x * N − λ, x * N + λ] with λ = 4.2ε ln N , following the suggestions in [11]. Outside this interval a coarse equidistant mesh is constructed.…”

“…Even if the layer-adapted mesh is correctly located (for our example, centered on x * N = x * = 0.5), nevertheless the computed solution can be inaccurate. In [11] it is proved that for a certain range of N and ε there exists a constantC such that…”

Analytic-Numerical Approach to Solving Singularly Perturbed Parabolic Equations with the Use of Dynamic Adapted Meshes

“…This idea has been used recently for problems with stationary interior layers in [1,2,3,4,5,6,7], where special grids have been used. In the case of moving interior layers, fairly complicated difference schemes has been constructed in the papers [2,8,9,10] and [11] where one example of periodic problem was considered.…”