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2 results

  • 21 - Stiff Systems of Ordinary Differential Equations and Linear Stability

  • from Part IV - Initial Value Problems for Ordinary Differential Equations
  • Abner J. Salgado, University of Tennessee, Knoxville, Steven M. Wise, University of Tennessee, Knoxville
  • Book:
    Classical Numerical Analysis
    Published online:
    29 September 2022
    Print publication:
    20 October 2022, pp 581-595

  • Summary

    The topic of this chapter is linear stability of schemes for ODEs. The notions of stiffness, linear stability domain, and A-stability are introduced. We discuss the A-stability of Runge-Kutta schemes via the amplification factor. The notion of A-acceptable, Pade approximations of the exponential, and the Hairer-Wanner-Norsett theorem are then presented. This allows us to show that all Gauss-Legendre-Runge-Kutta schemes are A-stable. For multistep schemes we present linear stability criteria and Dahlquist second barrier theorem. The boundary locus method concludes the chapter.

  • PRACTICAL RUNGE–KUTTA METHODS FOR SCIENTIFIC COMPUTATION

  • Part of
  • J. C. BUTCHER
  • Journal:
    The ANZIAM Journal / Volume 50 / Issue 3 / January 2009
    Published online by Cambridge University Press:
    03 November 2009, pp. 333-342
    • Article
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  • Implicit Runge–Kutta methods have a special role in the numerical solution of stiff problems, such as those found by applying the method of lines to the partial differential equations arising in physical modelling. Of particular interest in this paper are the high-order methods based on Gaussian quadrature and the efficiently implementable singly implicit methods.