A numerical method for a nonlinear singularly perturbed interior layer problem using an approximate layer location. (English) Zbl 1321.65121
Summary: A class of nonlinear singularly perturbed interior layer problems is examined in this paper. Solutions exhibit an interior layer at an a priori unknown location. A numerical method is presented that uses a piecewise uniform mesh refined around numerical approximations to successive terms of the asymptotic expansion of the interior layer location. The first term in the expansion is used exactly in the construction of the approximation which restricts the range of problem data considered. Denote the perturbation parameter as \(\varepsilon\) and the number of mesh intervals to be used as \(N\). The method is shown to converge point-wise to the true solution with a first order convergence rate (overlooking a logarithmic factor) for sufficiently small \(\varepsilon \leq N^{- 1}\). A numerical experiment is presented to demonstrate the convergence rate established.
MSC:
| 65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 34E15 | Singular perturbations for ordinary differential equations |
| 34B15 | Nonlinear boundary value problems for 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 |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
Keywords:
singularly perturbed problem; interior layer; nonlinear; Shishkin mesh; convergence; numerical experimentReferences:
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