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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 nth 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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