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Zbl 0983.65098
Ewing, Richard E.; Wang, Hong
A summary of numerical methods for time-dependent advection-dominated partial differential equations. (English)
[J] J. Comput. Appl. Math. 128, No.1-2, 423-445 (2001). ISSN 0377-0427

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 2000:: *65M06 Finite difference methods (IVP of PDE)
65-02 Research monographs (numerical analysis)
76S05 Flows in porous media
76M10 Finite element methods
35K15 Second order parabolic equations, initial value problems
65M60 Finite numerical methods (IVP of PDE)
76M20 Finite difference methods

Keywords: advection-diffusion equations; characteristic methods; survey paper; finite difference method; finite element method; discontinuous Galerkin method; porous medium fluid flow; Eulerian-Lagrangian methods

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Zentralblatt MATH Berlin [Germany]