Symmetrizers for Runge–Kutta Methods

N. Razali () and R. P. K. Chan ()
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N. Razali: Universiti Kebangsaan Malaysia, Fundamental Engineering Unit, Faculty of Engineering and Built Environment
R. P. K. Chan: University of Auckland, Department of Mathematics, Faculty of Science

A chapter in International Conference on Mathematical Sciences and Statistics 2013, 2014, pp 195-203 from Springer

Abstract: Abstract L-stable symmetrizers for symmetric Runge–Kutta methods are constructed to preserve the asymptotic error expansion in even powers of the stepsize and to provide necessary damping for the numerical solution of stiff initial value ordinary differential equations. The process is called symmetrization and has the effect of dampening down undesirable oscillations that may arise when the problem is stiff. The symmetry property can be exploited by Richardson extrapolation in increasing the order by two at a time. The effect of the damping is to stabilize the order behaviour of the extrapolation. In this paper, we discuss two types of symmetrizers; the well-known one-step smoothing formula of Gragg, and the new two-step smoothing formula. We construct these symmetrizers for the implicit midpoint and trapezoidal rules and investigate the active mode of application with extrapolation. Finally, we present numerical results in a constant stepsize setting that show two-step symmetrizers having certain advantages over one-step symmetrizers for stiff linear and nonlinear problems.

Keywords: Symmetric Runge-Kutta Methods; Asymptotic Error Expansion; Initial Value Ordinary Differential Equation; Richardson Extrapolation; Actual Extrapolation (search for similar items in EconPapers)
Date: 2014
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-981-4585-33-0_20

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DOI: 10.1007/978-981-4585-33-0_20

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