Runge-Kutta and MacCormack dynamics

David Griffiths, Desmond. J. Higham

    Research output: Working paper

    Abstract

    Our topic is the long term simulation of nonlinear differential equations, with a focus on spurious behaviour. We outline some of the recent work that has been done for initial value ordinary differential equations, and emphasize that finite, but spurious, solutions can arise for realistic choices of the timestep. As a concrete example, we exhibit spurious fixed points arising for the Improved Euler method with a logistic right-hand side. We then consider MacCormack's finite difference method applied to a linear advection equation with source term. From a linear stability perspective, due to of a lack of convexity in the stability region, instabilities may surface if the timestep is reduced while the Courant number is fixed at a realistic level. With a nonlinear source term, MacCormack's method admits spatially uniform spurious steady states that correspond to spurious fixed points of the Improved Euler method. We examine the stability of these spurious steady states for the logistic source term, and give numerical results.
    Original languageEnglish
    PublisherUniversity of Strathclyde
    Publication statusPublished - 1999

    Publication series

    Namein Proceedings of the 8th International Symposium on Computational Fluid Dynamics, Bremen, Sep. 1999
    PublisherUniversity of Strathclyde

    Keywords

    • Advection
    • Spurious solution
    • Stability
    • Timestepping

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  • Cite this

    Griffiths, D., & Higham, D. J. (1999). Runge-Kutta and MacCormack dynamics. (in Proceedings of the 8th International Symposium on Computational Fluid Dynamics, Bremen, Sep. 1999). University of Strathclyde.