On Implicit Runge-Kutta Methods with a Stability Function Having Distinct Real Poles
نویسنده
چکیده
Abstract. Because of their potential for offering a computational speed-up when used on certain multiprocessor computers, implicit Runge-Kutta methods with a stability function having distinct poles are analyzed. These are called multiply implicit (MIRK) methods, and because of the so-called order reduction phenomenon, their poles are required to be real, i.e., only real MIRK’s are considered. Specifically, it is proved that a necessary condition for a q-stage, real MIRK to be A-stable with maximal order q+1 is that q = 1, 2, 3, or 5. Nevertheless, it is shown that for every positive integer q, there exists a q-stage, real MIRK which is strongly A0-stable with order q + 1, and for every even q, there is a q-stage, real MIRK which is I-stable with order q. Finally, some useful examples of algebraically stable real MIRK’s are given.
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