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High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. (English) Zbl 1083.65514

Summary: In this paper we construct and analyze two compact monotone finite difference methods to solve singularly perturbed problems of convection-diffusion type. They are defined as HODIE methods of order two and three, i.e., the coefficients are determined by imposing that the local error be null on a polynomial space. For arbitrary meshes, these methods are not adequate for singularly perturbed problems, but using a Shishkin mesh we can prove that the methods are uniformly convergent of order two and three except for a logarithmic factor. Numerical examples support the theoretical results.

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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