An almost sixth-order finite-difference method for semilinear singular perturbation problems. (English) Zbl 1068.65103
The author deals with the application of finite difference schemes applied to singulary perturbed boundary value problems. The numerical methods use five-point formulae inside the boundary layer and lower-order formulae elsewhere. The schemes are of consistency order 6 with a constant being independent of the step size \(\delta\) and the singular perturbation parameter \(\varepsilon\). Moreover, under the condition \(\delta^2\leq\varepsilon\) a stability inequality and, thus, convergence is shown. Finally, some numerical examples are presented to show the real convergence order for fixed pairs of \(\varepsilon\) and \(\delta\) in various situations.
Reviewer: Johannes Schropp (Konstanz)
MSC:
65L12 | Finite difference and finite volume methods for ordinary differential equations |
65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
65L20 | Stability and convergence of numerical methods for ordinary differential equations |
34B15 | Nonlinear boundary value problems for ordinary differential equations |
34E15 | Singular perturbations for ordinary differential equations |