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Alternating direction numerical scheme for singularly perturbed 2D degenerate parabolic convection-diffusion problems. (English) Zbl 1427.65176

Summary: In this article, we study the numerical solution of singularly perturbed 2D degenerate parabolic convection-diffusion problems on a rectangular domain. The solution of this problem exhibits parabolic boundary layers along \(x = 0, y = 0\) and a corner layer in the neighborhood of \((0, 0)\). First, we use an alternating direction implicit finite difference scheme to discretize the time derivative of the continuous problem on a uniform mesh in the temporal direction. Then, to discretize the spatial derivatives of the resulting time semidiscrete problems, we apply the upwind finite difference scheme on a piecewise-uniform Shishkin mesh. We derive error estimate for the proposed numerical scheme, which shows that the scheme is \(\varepsilon \)-uniformly convergent of almost first-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
35B25 Singular perturbations in context of PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
65M22 Numerical solution of discretized equations 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
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