Twostep-by-twostep PIRK-type PC methods with continuous output formulas

Abstract

This paper deals with parallel predictor–corrector (PC) iteration methods based on collocation Runge–Kutta (RK) corrector methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. At n th step, the continuous output formulas are used not only for predicting the stage values in the PC iteration methods but also for calculating the step values at ( n + 2 ) th step. In this case, the integration processes can be proceeded twostep-by-twostep. The resulting twostep-by-twostep (TBT) parallel-iterated RK-type (PIRK-type) methods with continuous output formulas (twostep-by-twostep PIRKC methods or TBTPIRKC methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKC methods to a few widely-used test problems reveal that the new PC methods are much more efficient when compared with the well-known parallel-iterated RK methods (PIRK methods), parallel-iterated RK-type PC methods with continuous output formulas (PIRKC methods) and sequential explicit RK codes DOPRI5 and DOP853 available from the literature.