Nguyen Huu Cong

Explicit parallel two-step Runge-Kutta-Nyström methods

Abstract The aim of this paper is to design a class of two-step Runge-Kutta-Nystrom methods of arbitrarily high order for the special second-order equation y″(t) = f(y(t)), for use on parallel computers. Starting with an s-stage implicit two-step Runge-Kutta-Nystrom method of order p with k = p 2 implicit stages, we apply the highly parallel predictor-corrector iteration process in P(EC)mE mode. In this way, we obtain an explicit two-step Runge-Kutta-Nystrom method that has order p for all m and that requires k(m + 1) right-hand side evaluations per step of which each k evaluation can be computed in parallel. By a number of numerical experiments, we show the superiority of the parallel predictor-corrector methods proposed in this paper over both sequential and parallel methods available in the literature.

Numerical experiments with some explicit pseudo two-step RK methods on a shared memory computer☆

Abstract This paper investigates the performance of two explicit pseudo two-step Runge-Kutta methods of order 5 and 8 for first-order nonstiff ODEs on a parallel shared memory computer. For expensive right-hand sides the parallel implementation gives a speed-up of 3–4 with respect to the sequential one. Furthermore, we compare the codes with the two efficient nonstiff codes DOPRI5 and DOP853. For problems where the stepsize is determined by accuracy rather than by stability our codes are shown to be more efficient.

Note on the performance of direct and indirect Runge-Kutta-Nystro¨m methods

Abstract This paper deals with predictor-corrector iteration of Runge—Kutta—Nystrom (RKN) methods for integrating initial-value problems for special second-order ordinary differential equations. We consider RKN correctors based on both direct and indirect collocation techniques. The paper focuses on the convergence factors and stability regions of the iterated RKN correctors. It turns out that the methods based on direct collocation RKN correctors possess smaller convergence factors than those based on indirect collocation RKN correctors. Both families of methods have sufficiently large stability boundaries for nonstiff problems.

Parallel iteration of symmetric Runge-Kutta methods for nonstiff initial-value problems

Abstract This paper discusses parallel iteration schemes for collocation-based, symmetric Runge—Kutta (SRK) methods for solving nonstiff initial-value problems. Our main result is the derivation of four A-stable SRK corrector methods of orders 4, 6, 8 and 10 that optimize the rate of convergence when iterated by means of the highly parallel fixed-point iteration process. The resulting PISRK method (parallel iterated SRK method) shows considerably increased efficiency when compared with the fixed-point iteration process applied to Gauss—Legendre correctors.

Runge–Kutta–Nyström‐type parallel block predictor–corrector methods

This paper describes the construction of block predictor–corrector methods based on Runge–Kutta–Nyström correctors. Our approach is to apply the predictor–corrector method not only with stepsize h, but, in addition (and simultaneously) with stepsizes aih, i = 1 ...,r. In this way, at each step, a whole block of approximations to the exact solution at off‐step points is computed. In the next step, these approximations are used to obtain a high‐order predictor formula using Lagrange or Hermite interpolation. Since the block approximations at the off‐step points can be computed in parallel, the sequential costs of these block predictor–corrector methods are comparable with those of a conventional predictor–corrector method. Furthermore, by using Runge–Kutta–Nyström corrector methods, the computation of the approximation at each off‐step point is also highly parallel. Numerical comparisons on a shared memory computer show the efficiency of the methods for problems with expensive function evaluations.

RKN-type parallel block PC methods with Lagrange-type predictors☆

Abstract This paper describes the construction of block predictor-corrector methods based on Runge-Kutta-Nystrom correctors. Our approach is to apply the predictor-corrector method not only at step points, but also at off-step points (block points), so that in each step, a whole block of approximations to the exact solution at off-step points is computed. In the next step, these approximations are used to obtain a high-order predictor formula using Lagrange interpolation. By suitable choice of the abscissas of the off-step points, a much more accurately predicted value is obtained than by predictor formulas based on last step values. Since the block of approximations at the off-step points can be computed in parallel, the sequential costs of these block predictor-corrector methods are comparable with those of a conventional predictor-corrector method. Furthermore, by using Runge-Kutta-Nystrom corrector methods, the computation of the approximation at each off-step point is also highly parallel. Application of the resulting block predictor-corrector methods to a few widely-used test problems reveals that the sequential costs are reduced by a factor ranging from 4 to 50 when compared with the best sequential methods from the literature.

Explicit pseudo two-step runge-kutta methods for parallel computers ∗

The aim of this paper is to investigate a class of explicit pseudo two-step Runge-Kutta methods of arbitrarily high order for nonstiff problems for systems of first-order differential equations. By using collocation techniques we can obtain for any given order of accuracy p, a stable pth-order explicit pseudo two-step Runge-Kutta method requiring only one effective sequential right-hand side evaluation per step on multiprocessor computers. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature.

Continuous variable stepsize explicit pseudo two-step RK methods

Abstract The aim of this paper is to apply a class of constant stepsize explicit pseudo two-step Runge-Kutta methods of arbitrarily high order to nonstiff problems for systems of first-order differential equations with variable stepsize strategy. Embedded formulas are provided for giving a cheap error estimate used in stepsize control. Continuous approximation formulas are also considered for use in an eventual implementation of the methods with dense output. By a few widely used test problems, we compare the efficiency of two pseudo two-step Runge-Kutta methods of orders 5 and 8 with the codes DOPRI5, DOP853 and PIRK8. This comparison shows that in terms of ƒ-evaluations on a parallel computer, these two pseudo two-step Runge-Kutta methods are a factor ranging from 3 to 8 cheaper than DOPRI5, DOP853 and PIRK8. Even in a sequential implementation mode, fifth-order new method beats DOPRI5 by a factor more than 1.5 with stringent error tolerances.

A general class of explicit pseudo two-step RKN methods on parallel computers☆

Abstract The aim of this paper is to investigate a general class of explicit pseudo two-step Runge-Kutta-Nystrom methods (RKN methods) of arbitrarily high order for nonstiff problems for systems of special second-order differential equations y″( t ) = f(y( t )). Order and stability considerations show that we can obtain for any given p , a stable p th -order explicit pseudo two-step RKN method requiring p − 2 right-hand side evaluations per step of which each evaluation can be obtained in parallel. Consequently, on a multiprocessor computer, only one sequential right-hand side evaluation per step is required. By a few widely-used test problems, we show the superiority of the methods considered in this paper over both sequential and parallel methods available in the literature.

A parallel DIRK method for stiff initial-value problems

Abstract In this note we propose a fast parallel iteration process for solving a low-order implicit Runge–Kutta method. The resulting scheme can be regarded as a parallel singly diagonally implicit Runge–Kutta (PDIRK) method. On a two-processor computer, this method requires effectively the solution of two implicit relations per step. By two numerical experiments we compare this method with some sequential methods from the literature, and show its efficient behaviour.