A summary of numerical methods for time-dependent advection-dominated partial differential equations. (English) Zbl 0983.65098

Ewing, Richard E.; Wang, Hong

J. Comput. Appl. Math. 128, No. 1-2, 423-445 (2001).

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:

65M06	Finite difference methods for initial value and initial-boundary value problems involving PDEs
65-02	Research exposition (monographs, survey articles) pertaining to numerical analysis
76S05	Flows in porous media; filtration; seepage
76M10	Finite element methods applied to problems in fluid mechanics
35K15	Initial value problems for second-order parabolic equations
65M60	Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs
76M20	Finite difference methods applied to problems in fluid mechanics

Keywords:

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

Software:

SHASTA

References:

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