Compatible Fluxes for van Leer Advection

Application of the standard second-order upstream-centered advection method of van Leer to a coupled transport equation system of the form (∂/∂ t) [ρA]+∇·[ρA]u=0 can produce artificial extrema in the ratioT≡A/ρ, even thoughTsatisfies a simple advection equation and is expected to preserve monotonici...

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Bibliographic Details
Published in:Journal of computational physics 1998-10, Vol.146 (1), p.1-28
Main Authors: VanderHeyden, W.B., Kashiwa, B.A.
Format: Article
Language:English
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Summary:Application of the standard second-order upstream-centered advection method of van Leer to a coupled transport equation system of the form (∂/∂ t) [ρA]+∇·[ρA]u=0 can produce artificial extrema in the ratioT≡A/ρ, even thoughTsatisfies a simple advection equation and is expected to preserve monotonicity. Hereuis velocity, ρ is mass density, andAis a conserved quantity such as momentum density, energy density, or chemical species density. ThusTis a mass-specific transport quantity such as velocity, energy per unit mass, temperature, or species mass fraction. A new flux formulation and gradient limiting procedure is presented here which eliminates these artificial extrema, preserves the second-order accuracy, and preserves the monotone character of the van Leer method, even in the limit of vanishing mass density. Such a formulation is calledcompatible. The method is noniterative (i.e. explicit) and can be employed in a general finite-volume framework. Sample results for the transport of a square wave in one and two dimensions are provided.
ISSN:0021-9991
1090-2716
DOI:10.1006/jcph.1998.6070