Two-step-by-two-step PIRK-type PC methods based on Gauss–Legendre collocation points

This paper concerns with parallel predictor–corrector (PC) iteration methods for solving nonstiff initial-value problems (IVPs) for systems of first-order differential equations. The predictor methods are based on Adams-type formulas. The corrector methods are constructed by using coefficients of ss-stage collocation Gauss–Legendre Runge–Kutta (RK) methods based on c1,…,csc1,…,cs and the 2s2s-stage collocation RK methods based on c1,…,cs,1+c1,…,1+csc1,…,cs,1+c1,…,1+cs. At nnth integration step, the stage values of the 2s2s-stage collocation RK methods evaluated at tn+(1+c1)h,…,tn+(1+cs)htn+(1+c1)h,…,tn+(1+cs)h can be used as the stage values of the collocation Gauss–Legendre RK method for (n+2)(n+2)th integration step. By 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 RK-type (PIRK-type) PC methods based on Gauss–Legendre collocation points (two-step-by-two-step PIRKG methods or TBTPIRKG methods) give us a faster integration process. Fixed step size applications of these TBTPIRKG methods to the three widely used test problems reveal that the new parallel PC iteration methods are much more efficient when compared with the well-known parallel-iterated RK methods (PIRK methods) and sequential codes ODEX, DOPRI5 and DOP853 available from the literature.

► We propose and investigate new parallel integration methods for first-order ODEs. ► In these new methods, the integration is proceeded two-step-by-two-step. ► Compared with the best existing methods, the new methods are more efficient.