Twostep-by-twostep continuous PIRKN-type PC methods for nonstiff IVPs

Abstract

In this paper, we start with the consideration of direct collocation-based Runge–Kutta-Nyström (RKN) methods with continuous output formulas for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations y″( t) = f( t, y( t)). At nth step, the continuous output formulas can be used for calculating the step values at ( n + 2)th step and the integration processes can be proceeded twostep-by-twostep. In this case, we obtain twostep-by-twostep RKN methods with continuous output formulas (continuous TBTRKN methods). Furthermore, we consider a parallel predictor–corrector (PC) iteration scheme using the continuous TBTRKN methods as corrector methods with predictor methods defined by the continuous output formulas. The resulting twostep-by-twostep parallel-iterated RKN-type PC methods with continuous output formulas (twostep-by-twostep continuous PIRKN-type PC methods or TBTCPIRKN methods) give us a faster integration processes. Numerical comparisons based on the solution of a few widely-used test problems show that the new TBTCPIRKN methods are much more efficient than the well-known PIRKN methods, the famous nonstiff sequential ODEX2, DOP853 codes and comparable with the CPIRKN methods.