Báo cáo toán học: Efficiency of Embedded Explicit Pseudo Two-Step RKN Methods on a Shared Memory Parallel Computer

Mục đích của bài viết này là xây dựng hai nhúng rõ ràng giả hai bậc thang RKN phương pháp (nhúng EPTRKN phương pháp) trật tự 6 và 10 cho nonstiff initialvalue vấn đề (IVPs) y (t) = f (t, y (t)), y (t0) = y0, y (t0) = y0 và điều tra hiệu quả của họ trên các máy tính song song. Đối với hai phương pháp EPTRKN nhúng và các vấn đề tốn kém.
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Báo cáo toán học: " Efficiency of Embedded Explicit Pseudo Two-Step RKN Methods on a Shared Memory Parallel Computer"Vietnam Journal of Mathematics 34:1 (2006) 95–108 9LHWQD P -RXUQDO RI 0$ 7+ (0$ 7, &6 ‹ 9$67 Efficiency of Embedded Explicit Pseudo Two-Step RKN Methods on a Shared Memory Parallel Computer* N. H. Cong1 , H. Podhaisky2 , and R. Weiner2 1 Faculty of Math., Mech. and Inform., Hanoi University of Science 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam2 FB Mathematik und Informatik, Martin-Luther-Universit¨t Halle-Wittenberg a Theodor-Lieser-Str. 5, D-06120 Halle, Germany Received June 22, 2005Abstract. The aim of this paper is to construct two embedded explicit pseudo two-step RKN methods (embedded EPTRKN methods) of order 6 and 10 for nonstiff initial-value problems (IVPs) y (t) = f (t, y(t)), y(t0 ) = y0 , y (t0 ) = y0 and investigatetheir efficiency on parallel computers. For these two embedded EPTRKN methodsand for expensive problems, the parallel implementation on a shared memory parallelcomputer gives a good speed-up with respect to the sequential one. Furthermore, fornumerical comparisons, we solve three test problems taken from the literature by theembedded EPTRKN methods and the efficient nonstiff code ODEX2 running on thesame shared memory parallel computer. Comparing computing times for accuraciesreceived shows that the two new embedded EPTRKN methods are superior to the codeODEX2 for all the test problems.1. IntroductionThe arrival of parallel computers influences the development of numerical meth-ods for a nonstiff initial-value problem (IVP) for systems of special second-orderordinary differential equations (ODEs)∗ This work was supported by Vietnam NRPFS and the University of Halle.96 N. H. Cong, H. Podhaisky, and R. Weiner y , f ∈ Rd . y (t) = f (t, y(t)), y(t0 ) = y0 , y (t0 ) = y0 , (1.1)The most efficient numerical methods for solving this problem are the explicitRunge-Kutta-Nystr¨m (RKN) and extrapolation methods. In the literature, osequential explicit RKN methods up to order 11 can be found in e.g., [16-21,23, 28]. In order to exploit the facilities of parallel computers, a number ofparallel explicit methods have been investigated, for example in [2-6, 9-14]. Acommon challenge in the latter mentioned works is to reduce, for a given orderof accuracy, the required number of effective sequential f -evaluations per step,using parallel processors. In previous work of Cong et al. [14], a general class of explicit pseudo two-step RKN methods (EPTRKN methods) for solving problems of the form (1.1)has been investigated. These EPTRKN methods are ones of the cheapest parallelexplicit methods in terms of number of effective sequential f -evaluations per step.They can be easily equipped with embedded formulas for a variable stepsizeimplementation (cf. [9]). With respect to the number of effective sequential f -evaluations for a given accuracy, the EPTRKN methods have been shown to bemuch more efficient than most efficient sequential and parallel methods currentlyavailable for solving (1.1) (cf. [9, 14]). Most numerical comparisons of parallel and sequential methods are done bymeans of the number of effective sequential f -evaluations for a given accuracy ona sequential computer ignoring the communication time between processors (cf.e.g., 1, 3, 5, 6]). In comparisons of different codes running on parallel computers,the parallel codes often give disappointing results. However, in our recent work[15], two parallel codes EPTRK5 and EPTRK8 of oder 5 and 8, respectively,have been proposed. These codes are based on EPTRK methods consideredin [7, 8] which are a “first-order” version of the EPTRKN methods. The EP-TRK5 and EPTRK8 codes have been shown to be more efficient than the codesDOPRI5 and DOP853 for solving expensive nonstiff first-order problems on ashared memory parallel computer. We have also obtained a similar performanceof a parallel implementation of the BPIRKN codes for nonstiff special second-order problems (see [13]). These promising results encourage us to pursue theefficiency investigation of a real implementation of the EPTRKN methods on aparallel computer. This investigation consists of choosing relatively good embed-ded EPTRKN methods, defining reasonable error estimate for stepsi ...