An \(\varepsilon \)-uniform hybrid numerical scheme for a singularly perturbed degenerate parabolic convection-diffusion problem. (English) Zbl 1499.65414
Summary: In this paper, we study the numerical solution of singularly perturbed degenerate parabolic convection-diffusion problem on a rectangular domain. The solution of the problem exhibits a parabolic boundary layer in the neighbourhood of \(x=0\). First, we use the backward-Euler finite difference scheme to discretize the time derivative of the continuous problem on uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problem, we apply the hybrid finite difference scheme, which is a combination of central difference scheme and midpoint upwind scheme on piecewise uniform Shishkin mesh. We derive the error estimates, which show that the proposed hybrid scheme is \(\varepsilon \)-uniform convergent of almost second-order (up to a logarithmic factor) in space and first-order in time. Some numerical results have been carried out to validate the theoretical results.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65N06 | Finite difference methods for boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K65 | Degenerate parabolic equations |
Keywords:
singularly perturbed degenerate parabolic convection-diffusion problem; finite difference scheme; piecewise-uniform Shishkin meshes; error estimate; uniform convergenceReferences:
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