MacCormack's method for advection-reaction equations

David Griffiths, D. J. Higham

    Research output: Other contribution

    38 Downloads (Pure)

    Abstract

    MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic problems. Here, we consider MacCormack's method applied to the linear advection equation with nonlinear source term. Various features of the method are analysed. First, we show that the conventional implementation is not stable for Courant numbers close to one unless a small time-step is used. A simple modification, based on source term averaging, is shown to remove this defect. We then examine spurious fixed points that are inherited from the underlying Runge--Kutta method. Next we consider adapting the timestep as a means of improving the efficiency of the method. Theoretical analysis based on the method of modified equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be refined. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modified error measure that takes account of immanent spatial errors.
    Original languageEnglish
    TypeTechnical report
    Media of outputText
    Publication statusUnpublished - 2001

    Fingerprint

    Advection
    Source Terms
    Advection Equation
    Nonlinear Source
    Hyperbolic Problems
    Error Control
    Modified Equations
    Runge-Kutta Methods
    Finite Difference Scheme
    Traveling Wave
    Averaging
    Linear equation
    Theoretical Analysis
    Defects
    Fixed point

    Keywords

    • Advection-reaction equations
    • Mathematics
    • Fluid dynamics

    Cite this

    Griffiths, D., & Higham, D. J. (2001). MacCormack's method for advection-reaction equations. Unpublished.
    Griffiths, David ; Higham, D. J. / MacCormack's method for advection-reaction equations. 2001.
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    abstract = "MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic problems. Here, we consider MacCormack's method applied to the linear advection equation with nonlinear source term. Various features of the method are analysed. First, we show that the conventional implementation is not stable for Courant numbers close to one unless a small time-step is used. A simple modification, based on source term averaging, is shown to remove this defect. We then examine spurious fixed points that are inherited from the underlying Runge--Kutta method. Next we consider adapting the timestep as a means of improving the efficiency of the method. Theoretical analysis based on the method of modified equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be refined. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modified error measure that takes account of immanent spatial errors.",
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    MacCormack's method for advection-reaction equations. / Griffiths, David; Higham, D. J.

    2001, Technical report.

    Research output: Other contribution

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    AU - Griffiths, David

    AU - Higham, D. J.

    N1 - dc.ispartof: Strathclyde University Mathematics Research Report;25 dc.type: Article Article

    PY - 2001

    Y1 - 2001

    N2 - MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic problems. Here, we consider MacCormack's method applied to the linear advection equation with nonlinear source term. Various features of the method are analysed. First, we show that the conventional implementation is not stable for Courant numbers close to one unless a small time-step is used. A simple modification, based on source term averaging, is shown to remove this defect. We then examine spurious fixed points that are inherited from the underlying Runge--Kutta method. Next we consider adapting the timestep as a means of improving the efficiency of the method. Theoretical analysis based on the method of modified equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be refined. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modified error measure that takes account of immanent spatial errors.

    AB - MacCormack's method is an explicit, second order finite difference scheme that is widely used in the solution of hyperbolic problems. Here, we consider MacCormack's method applied to the linear advection equation with nonlinear source term. Various features of the method are analysed. First, we show that the conventional implementation is not stable for Courant numbers close to one unless a small time-step is used. A simple modification, based on source term averaging, is shown to remove this defect. We then examine spurious fixed points that are inherited from the underlying Runge--Kutta method. Next we consider adapting the timestep as a means of improving the efficiency of the method. Theoretical analysis based on the method of modified equations is combined with numerical tests on a travelling wave problem in order to give a feel for how the time-step should be refined. An adaptive approach based on temporal local error control is shown to have serious drawbacks. Much better performance is obtained with a modified error measure that takes account of immanent spatial errors.

    KW - Advection-reaction equations

    KW - Mathematics

    KW - Fluid dynamics

    M3 - Other contribution

    ER -