A positive splitting method for mixed hyperbolic-parabolic systems

Alf Gerisch, David F. Griffiths, Rudiger Weiner, Mark A. J. Chaplain

    Research output: Contribution to journalArticle

    26 Citations (Scopus)

    Abstract

    In this article we present a method of lines approach to the numerical solution of a system of coupled hyperbolic - parabolic partial differential equations (PDEs). Special attention is paid to preserving the positivity of the solution of the PDEs when this solution is approximated numerically. This is achieved by using a flux-limited spatial discretization for the hyperbolic equation. We use splitting techniques for the solution of the resulting large system of stiff ordinary differential equations. The performance of the approach applied to a biomathematical model is compared with the performance of standard methods.
    Original languageEnglish
    Pages (from-to)152-168
    Number of pages17
    JournalNumerical Methods for Partial Differential Equations
    Volume17
    Issue number2
    DOIs
    Publication statusPublished - 2001

    Fingerprint

    Splitting Method
    Parabolic Systems
    Hyperbolic Systems
    Partial differential equations
    Stiff Ordinary Differential Equations
    Method of Lines
    Hyperbolic Partial Differential Equations
    Parabolic Partial Differential Equations
    Hyperbolic Equations
    Ordinary differential equations
    Positivity
    Partial differential equation
    Discretization
    Numerical Solution
    Fluxes
    Model
    Standards

    Keywords

    • Mixed hyperbolic-parabolic PDE system
    • Finite difference approximation
    • Method of lines
    • Splitting methods
    • Positive methods

    Cite this

    Gerisch, Alf ; Griffiths, David F. ; Weiner, Rudiger ; Chaplain, Mark A. J. / A positive splitting method for mixed hyperbolic-parabolic systems. In: Numerical Methods for Partial Differential Equations. 2001 ; Vol. 17, No. 2. pp. 152-168.
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    keywords = "Mixed hyperbolic-parabolic PDE system, Finite difference approximation, Method of lines, Splitting methods, Positive methods",
    author = "Alf Gerisch and Griffiths, {David F.} and Rudiger Weiner and Chaplain, {Mark A. J.}",
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    A positive splitting method for mixed hyperbolic-parabolic systems. / Gerisch, Alf; Griffiths, David F.; Weiner, Rudiger; Chaplain, Mark A. J.

    In: Numerical Methods for Partial Differential Equations, Vol. 17, No. 2, 2001, p. 152-168.

    Research output: Contribution to journalArticle

    TY - JOUR

    T1 - A positive splitting method for mixed hyperbolic-parabolic systems

    AU - Gerisch, Alf

    AU - Griffiths, David F.

    AU - Weiner, Rudiger

    AU - Chaplain, Mark A. J.

    N1 - dc.publisher: Wiley-Blackwell

    PY - 2001

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    AB - In this article we present a method of lines approach to the numerical solution of a system of coupled hyperbolic - parabolic partial differential equations (PDEs). Special attention is paid to preserving the positivity of the solution of the PDEs when this solution is approximated numerically. This is achieved by using a flux-limited spatial discretization for the hyperbolic equation. We use splitting techniques for the solution of the resulting large system of stiff ordinary differential equations. The performance of the approach applied to a biomathematical model is compared with the performance of standard methods.

    KW - Mixed hyperbolic-parabolic PDE system

    KW - Finite difference approximation

    KW - Method of lines

    KW - Splitting methods

    KW - Positive methods

    U2 - 10.1002/1098-2426(200103)17:2<152::AID-NUM5>3.0.CO;2-A

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