A block 6(4) Runge-Kutta formula for nonstiff initial value problems

Abstract

A new selection is made for an efficient two-step block Runge-Kutta formula of order 6. The new formula is developed using some of the efficiency criteria recently investigated by Shampine, and as a result, a block formula with much improved performance is obtained. An important property of this new formula is that there is a "natural" interpolating polynomial available. This can be used to compute approximate solution values at off-step points without the need to compute any additional function evaluations. The quality of this interpolant is examined, and it is shown to have certain desirable properties. The performance of the new block Runge-Kutta formula is evaluated using the DETEST test set and is shown to be more efficient than certain other standard Runge-Kutta formulas for this particular test set.

Global Error versus Tolerance for Explicit Runge-Kutta Methods

“…In addition to making an interpolant q(t) available, it is common for modern initial value codes to provide access to q'{t) (see, for example, Cash, 1989). In this section we look at the behaviour of the corresponding global error, From Corollary 2.1, we see that if a suitable error control strategy is used, the ideal interpolant r\i(t) satisfies condition A of Theorem 2.1 and hence where v'(t) is independent of 6 and continuous.…”

Parallel defect control

“…), p,(x~+~) = Y~+l andp~,(x~+t) = f(x.+l, Yn÷l), so that the corresponding piecewise approximation is globally C 1. Many such interpolation schemes have been derived recently, largely in conjunction with low order Runge-Kutta formulas (see, for example, [3,5,8,12,15,21,23]). The scheme that we propose below falls into the class defined by Shampine [22]; see also Gladwell [11] for a related approach.…”

Runge–Kutta Defect Control Using Hermite–Birkhoff Interpolation

“…These methods produce discrete approximations y, y(x,) by proceeding in a stepwise fashion; a typical step involves advancing the numerical approximation from x, to X,+l :x, + h,. To complement the approximation at the meshpoints {x,}, many authors have derived interpolants p(x) which provide approximations p(x) y(x) for other values of x (see, for example, 1 ], [7], [9], [ 12], 13]). It is desirable for p(x) to provide efficient, accurate approximations, and to have at least global C continuity.…”

“…If the problem is to be solved by performing a grid search, then the lemma allows us to reduce the amount of searching. For example, if r 5, then rather than considering y3, 3'4, ye (0, 1), we may restrict attention to ys (0, 1) and 73, Y4 (0,1/2]. (We may also assume, without loss of generality, the ordering…”

“…y=2Xyllog(max(y2,10-3)), yl (1) exp (sin (1)), y= -2xy2 log (max (Yl, 10-3)), y2(1) exp (cos (1)), 1 _-<x_<-5. with values of 0.1, 0.5, and 0.9, respectively, for the eccentricity parameter e. The results can be found in Tables 1 and 2. We see that for the Hermite-Birkhoff scheme, the defect sample is generally very reliable, more so than for the original robust schemes in 10]. It is noticeable that the How to compare numerically two different defect control schemes, using such criteria as efficiency, reliability, and the proportionality of the global error to the tolerance.…”