A class of high-order Runge-Kutta-Chebyshev stability polynomials

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    Abstract

    The analytic form of a new class of factorized Runge-Kutta-Chebyshev (FRKC) stability polynomials of arbitrary order N is presented. Roots of FRKC stability polynomials of degree L=MN are used to construct explicit schemes comprising L forward Euler stages with internal stability ensured through a sequencing algorithm which limits the internal amplification factors to ~L2. The associated stability domain scales as M2 along the real axis. Marginally stable real-valued points on the interior of the stability domain are removed via a prescribed damping procedure.By construction, FRKC schemes meet all linear order conditions; for nonlinear problems at orders above 2, complex splitting or Butcher series composition methods are required. Linear order conditions of the FRKC stability polynomials are verified at orders 2, 4, and 6 in numerical experiments. Comparative studies with existing methods show the second-order unsplit FRKC2 scheme and higher order (4 and 6) split FRKCs schemes are efficient for large moderately stiff problems.

    Original languageEnglish
    Pages (from-to)665-678
    Number of pages14
    JournalJournal of Computational Physics
    Volume300
    DOIs
    Publication statusPublished - 2015

    Keywords

    • Method of lines
    • Stability and convergence of numerical methods
    • Stiff equations

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