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On piecewise-uniform meshes for upwind- and central-difference operators for solving singularly perturbed problems. (English) Zbl 0814.65082

This paper deals with the ordinary differential equation \(\varepsilon u_{xx}+ u_ x= 0\) on (0,1), \(\varepsilon> 0\) with boundary conditions \(u(0)= 0\) and \(u(1)= 1\). It is well known that central-difference operators on uniform meshes, applied to singularly perturbed problems can produce approximate solutions having oscillations that are unbounded when \(\varepsilon\to 0\), and even the use of upwind-difference operators on uniform meshes does not necessarily give satisfactory numerical solutions in practice.
It is shown, with specific analytic examples, that both upwind- and central-difference operators on specially designed piecewise-uniform meshes give numerical methods which do not suffer from this defect. Conditions are also given on the structure of a piecewise uniform mesh that are necessary if the numerical method, composed of this mesh and an upwind-difference operator, is to be convergent uniformly with respect to the singular perturbation parameter.

MSC:

65L10 Numerical solution of boundary value problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
65L50 Mesh generation, refinement, and adaptive methods for ordinary differential equations
34B05 Linear boundary value problems for ordinary differential equations
34E15 Singular perturbations for ordinary differential equations
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