Reformulations for general advection–diffusion–reaction equations and locally implicit ADER schemes

Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection–diffusion–reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a...

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Bibliographic Details
Published inJournal of computational physics Vol. 275; pp. 415 - 442
Main Authors Montecinos, Gino I., Toro, Eleuterio F.
Format Journal Article
LanguageEnglish
Published Elsevier Inc 15.10.2014
Subjects
ADER high-order schemes
Advection–diffusion–reaction equations
Balancing
Cauchy–Kowalewski procedure
Compressibility
Dynamical systems
Generalised Riemann problems
Ingredients
Mathematical analysis
Mathematical models
Navier-Stokes equations
Nonlinearity
Relaxation methods
ADER high-order schemes
Relaxation methods
Generalised Riemann problems
Advection–diffusion–reaction equations
Cauchy–Kowalewski procedure
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Summary:Following Cattaneo's original idea, in this article we first present two relaxation formulations for time-dependent, non-linear systems of advection–diffusion–reaction equations. Such formulations yield time-dependent non-linear hyperbolic balance laws with stiff source terms. Then we present a locally implicit version of the ADER method to solve these stiff systems to high accuracy. The new ingredient of the numerical methodology is a locally implicit solution of the generalised Riemann problem. We illustrate the formulations and the resulting numerical approach by solving the compressible Navier–Stokes equations.
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ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2014.06.018