Solution of ordinary differential initial value problems on an intel hypercube

Abstract

~Schemes for the solution of linear initial or boundary value problenm on a hypercube were developed by Katti and Neta [1] and tested and improved by Lustman, Neta and Katti [2]. Amov.g other procedures for parallel computers, fully implicit Runge-Kutta methods were discussed by Jackson and Norsett [3] and Lie [4]. Here, we develop a method based on extrapolation to the limit, which is useful even for nonlinear problesms. Numerical experiments show excellent accuracy when low order schemes are combined with polynomial extrapolation.

Scalability and locality of extrapolation methods on large parallel systems

“…Thus, the absolute runtimes on JUROPA are smaller than those on HLRB 2. Because of the similar cache sizes, the implementations on JUROPA run out of caches at similar system sizes as those on HLRB 2: Two vectors of n components fit in the Xeon's L2 cache for system sizes of up to n ≈ 1.6×10 4 and in the L3 cache for system sizes of up to n ≈ 5×10 5 . Figure 14 shows the corresponding increases in the runtime of the general implementation.…”

“…Figure 14). In the two parallel implementations, for 25 threads, each thread is responsible for about 3.2×10 5 Copyright components. For this number of components, the runtimes of the sequential implementations do not differ significantly.…”

“…For larger system sizes, the runtimes rapidly increase, especially for the general implementation. The L3 cache can hold two vectors of n components for system sizes of up to n ≈ 5.9×10 5 . At this point, a steep increase is observed for the runtimes of the general implementation.…”

Parallel, adaptive, multi-object trajectory integrator for space simulation applications

“…Investigations related to development of different methods (Fehlberg, 1968(Fehlberg, , 1969Dormand and Prince, 1978;Bettis, 1978;Lustman et al, 1991Lustman et al, , 1992 have been carried and their application in the field of space dynamics (Sharp, 1991(Sharp, , 2003 has been studied. Many computer programs based on different integration methods (integrators, solvers) have been reported (Van Der Houwen and Sommeijer, 1990;Byrne and Hindmarsh, 1999;Petcu, 2004).…”

Scalable Parallel RK Solvers for ODEs Derived by the Method of Lines

“…These include extrapolation methods [8], relaxation techniques [2], multiple shooting [6], and iterated RungeKutta methods, which are predictor-corrector methods based on implicit Runge-Kutta methods [9,14]. Two-step RK methods based on the computation of s stage approximations are proposed in [10].…”