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Finite difference scheme for parabolic problems on composite grids with refinement in time and space. (English) Zbl 0817.65071
Authors’ abstract: Finite difference schemes for transient convection- diffusion problems on grids with local refinement in time and space are constructed and studied. The construction utilizes a modified upwind approximation and linear interpolation at the slave nodes. The proposed schemes are implicit of the backward Euler type and are unconditionally stable. Error analysis in the maximum norm is presented, and convergence estimates for smooth solutions are derived. Optimal approximation results are shown for ratios between the spatial and time discretization parameters away from the CFL condition. Finally, numerical examples illustrating the theory are given.
MSC:
65M06Finite difference methods (IVP of PDE)
65M12Stability and convergence of numerical methods (IVP of PDE)
65M15Error bounds (IVP of PDE)
65M50Mesh generation and refinement (IVP of PDE)
35K15Second order parabolic equations, initial value problems