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A summary of numerical methods for time-dependent advection-dominated partial differential equations. (English) Zbl 0983.65098
Summary: We give a brief summary of numerical methods for time-dependent advection-dominated partial differential equations (PDEs), including first-order hyperbolic PDEs and nonstationary advection-diffusion PDEs. Mathematical models arising in porous medium fluid flow are presented to motivate these equations. It is understood that these PDEs also arise in many other important fields and that the numerical methods reviewed apply to general advection-dominated PDEs. We conduct a brief historical review of classical numerical methods, and a survey of the recent developments on the Eulerian and characteristic methods for time-dependent advection-dominated PDEs. The survey is not comprehensive due to the limitation of its length, and a large portion of the paper covers characteristic or Eulerian-Lagrangian methods.

MSC:
65M06Finite difference methods (IVP of PDE)
65-02Research monographs (numerical analysis)
76S05Flows in porous media; filtration; seepage
76M10Finite element methods (fluid mechanics)
35K15Second order parabolic equations, initial value problems
65M60Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE)
76M20Finite difference methods (fluid mechanics)
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References:
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