Richardson extrapolation technique for singularly perturbed parabolic convection–diffusion problems. (English) Zbl 1243.65106
This paper deals with the study of a post-processing technique for one-dimensional singularly perturbed parabolic convection-diffusion problems exhibiting a regular boundary layer. For discretizing the time derivative, we use the classical backward-Euler method and for the spatial discretization the simple upwind scheme is used on a piecewise-uniform Shishkin mesh. We show that the use of Richardson extrapolation technique improves the \(\varepsilon \)-uniform accuracy of simple upwinding in the discrete supremum norm from \(O (N ^{-1} \ln N + \varDelta t)\) to \(O (N ^{ - 2} \ln ^{2} N + \varDelta t ^{2})\), where \(N\) is the number of mesh-intervals in the spatial direction and \(\Delta t\) is the step size in the temporal direction. The theoretical result is also verified computationally by applying the proposed technique on two test examples.
Reviewer: Muhammad Akram (Lahore)
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 35B25 | Singular perturbations in context of PDEs |
| 35K20 | Initial-boundary value problems for second-order parabolic equations |
Keywords:
singular perturbation; regular boundary layer; upwind scheme; Richardson extrapolation; piecewise-uniform Shishkin mesh; uniform convergence; numerical examples; parabolic convection-diffusion problems; backward-Euler methodReferences:
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