Explicit, high-order Runge-Kutta-Nyström methods for parallel computers

Abstract

Sommeijer, B.P., Explicit, high-order Runge-Kutta-Nystrom methods for parallel computers, Applied Numerical Mathematics 13 (1993) 221-240.The paper describes the construction of explicit Runge-Kutta-Nystri:im (RKN) methods of arbitrarily high order. The order is borrowed from an underlying implicit RKN method. For the approximate solution of this method, an iteration scheme is defined. Prescribing a fixed number of iterations, the resulting scheme is an explicit RKN method. The iteration scheme is defined in such a way that many of the right-hand side evaluations can be done concurrently. As a result, explicit RKN schemes of order p are obtained which require, on a parallel computer, approximately p /2 right-hand side evaluations per step. Both in fixed-and variable-step mode, the schemes are compared with existing (sequential) high-order RKN methods from the literature and are shown to demonstrate superior behaviour.

An algorithm for second order initial and boundary value problems with an automatic error estimate based on a third derivative method

“…The results obtained using the H8 and BG8 methods are reproduced in Table 5 and compared to the results given by the TDM. It is seen from Table 5 that for approximately the same FNCs our method performs better than those in [24] in terms of accuracy (larger CD values), despite the fact that those methods are of a higher order. In Table 6, we show that the calculated ROC of the TDM is consistent with the theoretical order ( p = 6) behavior of the method, since on halving the step size, Err is reduced by a factor of about 2 6 .…”

“…The problem was also solved in [24] using the eighth-order, eight-stage RKN (H8) method constructed by Hairer [14] and the eighth-order, nine-stage method (BG8) constructed by Beentjes Gerritsen [5]. The maximum norm of the global error for the ycomponent is given in the form 10 −CD , where CD denotes the number correct decimal digits at the endpoint (see [24]). We have chosen to compare these methods of order 8 with the TDM of order 6, because the orders of the methods are moderately close.…”

Block third derivative method based on trigonometric polynomials for periodic initial-value problems

“…Fig. 1 The stability region for the TFBTDM plotted in the (q, u)-plane Definition 2.5 As u → 0, the interval [−q 0 , 0] is the stability interval, if in this interval ρ(M(q, 0)) ≤ 1 and q 0 is the stability boundary (see [27]). …”

“…The linear-stability of the TFBTDM is discussed by applying the method to the test equation y = λy, where λ is expected to run through the (negative) eigenvalues of the Jacobian matrix ∂ f ∂ y (see [27]). Letting q = λh 2 , it is easily shown that the application of (12) to the test equation yields (13) where the matrix M(q, u) is the amplification matrix which determines the stability of the method.…”

“…It has been shown that solving (1) directly is preferable, since about half of the storage space can be saved, especially, if the dimension of (1) is large (see [4,6,11,[11][12][13]20,25,[27][28][29]). Nevertheless, fewer methods have been proposed for solving the general form (2) directly (see [1,3,12,34]).…”