Generalized symmetric Runge-Kutta methods

Abstract

In this paper the concept of symmetry for Runge-Kutta methods is generalized to include composite methods. The extrapolations of the usual compositions of a symmetric method ℛ of the form Open image in new window are shown not to beA-stable. However, this limitation can be overcome by considering composite methods of the form Open image in new window where Open image in new window represents a non-symmetric and possiblyL-stable method called a symmetrizer satisfying Open image in new window . While no longer symmetric, these composite methods yet satisfy Open image in new window and thus share with symmetric methods the important property of admitting asymptotic error expansions in even powers of 1/n. Composite methods that are constructed in this way and presented in this paper have implementation costs comparable to that for the symmetric method. They generalize those based on the implicit midpoint and trapezoidal rules used with the standard smoothing formulae and thus extend the class of methods for acceleration techniques of extrapolation and defect correction. A characterization ofL-stable symmetrizers for 2-stage symmetric methods is given and studied for a particular stiff model problem. The analysis suggests that certainL-stable symmetrizers can play an important role in suppressing order defect effects for stiff problems.