Attainable Orders of Phase and Amplification Error of Explicit Parallel Runge-Kutta Methods

Abstract

In this paper, we study the orders of the two errors, phase and amplification errors, of explicit parallel Runge-Kutta methods for the integrations of periodic initial value problems. We first give a mathematical definition of the effective stages of the parallel Runge-Kutta method. Using the result, we show that the orders of these two errors depend on the number of the effective stages in parallel executions of the methods, and that the sum of these orders is bounded by 2s +1, where s is the number of the effective stages.