Robust difference scheme for the Cauchy problem for a singularly perturbed ordinary differential equation

Abstract

Grid approximation of the Cauchy problem on the interval  D = {0 ≤ x ≤ d} is first studied for a linear singularly perturbed ordinary differential equation of the first order with a perturbation parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval (0, 1]. In the Cauchy problem under consideration, for small values of the parameter ε, a boundary layer of width O(ε) appears on which the solution varies by a finite value. It is shown that, for such a Cauchy problem, the solution of the standard difference scheme on a uniform grid does not converge ε-uniformly in the maximum norm; convergence occurs only under the condition h ε, where h = d N ⁻¹ , N is the number of grid intervals, h is the grid step-size. Taking into account the behavior of the singular component in the solution, a special piecewise-uniform grid is constructed that condenses in a neighborhood of the boundary layer. It is established that the standard difference scheme on such a special grid converges ε-uniformly in the maximum norm at the rate O(N ⁻¹ lnN). Such a scheme is called a robust one.

For a model Cauchy problem for a singularly perturbed ordinary differential equation, standard difference schemes on a uniform grid (a classical difference scheme) and on a piecewise-uniform grid (a special difference scheme) are constructed and investigated. The results of numerical experiments are given, which are consistent with theoretical results.

Keyword : singularly perturbed Cauchy problem, ordinary differential equation, boundary layer, a priori estimates, standard difference scheme, uniform grid, piecewise-uniform grid, maximum norm, solution decomposition, robust difference scheme