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Front tracking and operator splitting for nonlinear degenerate convection-diffusion equations. (English) Zbl 0961.65073
Bjørstad, Petter (ed.) et al., Parallel solution of partial differential equations. Proceedings of a workshop, Univ. of Minnesota, Minneapolis, MN, USA, June 9-13, 1997. New York, NY: Springer. IMA Vol. Math. Appl. 120, 209-227 (2000).
Summary: We describe two variants of an operator splitting strategy for nonlinear, possibly strongly degenerate convection-diffusion equations. The strategy is based on splitting the equations into a hyperbolic conservation law for convection and a possibly degenerate parabolic equation for diffusion. The conservation law is solved by a front tracking method, while the diffusion equation is here solved by a finite difference scheme. The numerical methods are unconditionally stable in the sense that the (splitting) time step is not restricted by the spatial discretization parameter. The strategy is designed to handle all combinations of convection and diffusion (including the purely hyperbolic case). Two numerical examples are presented to highlight the features of the methods, and the potential for parallel implementation is discussed.
For the entire collection see [Zbl 0938.00013].

MSC:
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
35K65 Degenerate parabolic equations
35L80 Degenerate hyperbolic equations
65M12 Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs
35K55 Nonlinear parabolic equations
35L70 Second-order nonlinear hyperbolic equations
35M10 PDEs of mixed type
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