A uniformly convergent scheme for two-parameter problems having layer behaviour. (English) Zbl 1499.65325
Summary: We present a numerical scheme for the solution of two-parameter singularly perturbed problems whose solution has multi-scale behaviour in the sense that there are small regions where the solution changes very rapidly (known as layer regions) otherwise the solution is smooth (known as a regular region) throughout the domain of consideration. In particular, to solve the problems whose solution exhibits twin boundary layers at both endpoints of the domain of consideration, we propose a collocation method based on the quintic \(\mathcal{B}\)-spline basis functions. A piecewise-uniform mesh that increases the density within the layer region compared to the outer region is used. An \((N+1) \times (N+1)\) penta-diagonal system of algebraic equations is obtained after the discretization. A well-known fast penta-diagonal system solver algorithm is used to solve the system. We have shown that the method is almost fourth-order parameters uniformly convergent. The theoretical estimates are verified through numerical simulations for two test problems.
MSC:
| 65L10 | Numerical solution of boundary value problems involving ordinary differential equations |
| 65L11 | Numerical solution of singularly perturbed problems involving ordinary differential equations |
| 65L20 | Stability and convergence of numerical methods for ordinary differential equations |
| 65L50 | Mesh generation, refinement, and adaptive methods for ordinary differential equations |
| 65L70 | Error bounds for numerical methods for ordinary differential equations |
Keywords:
singularly perturbed problems; two-parameter; \(\mathcal{B}\)-splines; collocation method; Shishkin mesh; boundary layerReferences:
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