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.
Highlights
The Cauchy problem for regular equations was investigated, e.g., in well known works [3, 4, 5, 7, 9, 12], but for singularly perturbed equations it has not been considered previously
We consider a new class of singularly perturbed problems, namely, the Cauchy problem for a linear singularly perturbed ordinary first-order differential equation with a small parameter ε multiplying the derivative in the equation where the parameter ε takes arbitrary values in the half-open interval
It is shown that the use of standard difference schemes on uniform grids leads to large errors of the grid solutions that makes these classical difference schemes unsuitable for practical calculations
Summary
The Cauchy problem for regular equations was investigated, e.g., in well known works [3, 4, 5, 7, 9, 12], but for singularly perturbed equations it has not been considered previously. For wide classes of singularly perturbed boundary value problems with elliptic and ordinary differential equations, and for initial boundary value problems with parabolic equations, special numerical methods based on standard difference schemes on grids condensing in boundary layers have been developed and well studied whose solutions converge ε-uniformly in the maximum norm (see, e.g., [1, 2, 6, 10, 11] and the bibliography therein). Our aim is for Cauchy problem (2.2), (2.1) for a linear singularly perturbed ordinary differential equation, to study applicability of standard difference schemes for solving such problems; to construct and investigate a special difference scheme convergent ε-uniformly in the maximum norm, i.e., robust difference scheme. Consider a standard difference scheme for Cauchy problem (2.2), (2.1) constructed on the basis of a monotone grid approximation of a differential equation (see, e.g., [9] in the case of a regular equation)
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