Two-step-by-two-step PIRKN-type PC methods based on Gauss–Legendre collocation points for nonstiff IVPs

Abstract

This paper investigates parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of special second-order differential equations of the form \(\mathbf{y}''(t)=\mathbf{f}(t,\mathbf{y}(t))\). The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of \(s\)-stage collocation Gauss–Legendre Runge–Kutta–Nystrom (RKN) methods based on \(\hat{c}_1,\dots ,\hat{c}_s\) and \(2s\)-stage collocation RKN methods based on \(\hat{c}_1,\dots ,\hat{c}_s,1+\hat{c}_1,\dots ,1+\hat{c}_s\). At \(n\)th integration step, the stage values of the \(2s\)-stage collocation RKN methods evaluated at \(t_n+(1+\hat{c}_1)h,\dots ,t_n+(1+\hat{c}_s)h\) can be used as the stage values of the collocation Gauss–Legendre RKN method for \((n+2)\)th integration step. In this way we obtain the corrector methods in which the integration processes can be proceeded two-step by two-step. The resulting parallel PC iteration methods which are called two-step-by-two-step (TBT) parallel-iterated RKN-type (PIRKN-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKNG methods or TBTPIRKNG methods) give us a faster integration process. Fixed stepsize applications of these TBTPIRKNG methods to three widely used test problems show that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RKN methods (PIRKN methods) and sequential codes ODEX2 and DOP853 available from the literature.