Clavero, C.; Gracia, J. L.; Lisbona, F. High order methods on Shishkin meshes for singular perturbation problems of convection-diffusion type. (English) Zbl 1083.65514 Numer. Algorithms 22, No. 1, 73-97 (1999). 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. Cited in 24 Documents 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 Keywords:singular perturbation; uniform convergence; Shishkin mesh; high order PDFBibTeX XMLCite \textit{C. Clavero} et al., Numer. Algorithms 22, No. 1, 73--97 (1999; Zbl 1083.65514) Full Text: DOI