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Vietnam Journal of Mathematics 35:2(2007) 223-229  Twostep-by-twostep PIRKC Methods Nguyen Huu Cong and Le Ngoc Xuan Abstract.  This paper concerns 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) PC methods with continuous output formulas (twostep-by-twostep PIRKC methods or TBTPIRKC methods) paralled-iterated RK-type PC methos with continuous output for mulas (PIRKC method) give us a faster integration process. 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.   2000 Mathematics Subject Classification: 65L05, 65L06. Keywords: Runge-Kutta methods, predictor-corrector methods, stability, parallelism.

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