Krogh, F. T. "Changing Stepsize in the Integration of Differential Equations using Modified Divided Differences." Lectures notes in Mathematics 362 pp22-71, Springer-Verlag (1973).  Home/Search   Document Not in Database   Summary   Related Articles   Check

....to the initial singularity, a; Q a ) 0; 0) is not suitable as a starting point. Alternative starting values (i.e. initial conditions) should be taken either from the equation (3) or from the integration coefficients (typically obtained by recurrence relations) computed by the host ODE code [10, 12] to advance the solution. In this latter case, such choices could be based on the stepsize from the last step or a value for the projected step obtained by a standard stepsize estimator. Since stepsize control requires typically only a very crude approximation to h , it could be obtained using ....

....reasons: ffl they provide a simple numerical approach to an otherwise difficult polynomial rootfinding problem and ffl their accuracy can be tailored to the accuracy required by the stepsize strategy. Expensive full precision evaluations of Q(h) using recurrence relations of the type outlined in [10, 12] are thus avoided. The direct solution of (4) in the Adams EPS case is, however, complicated by the singularity at h = 0. As we now show, this does not occur for EPUS schemes: they are singularity free. Theoretically, it is hoped that equations of the form (4) may also provide insight into how new ....

Krogh, F. T. "Changing Stepsize in the Integration of Differential Equations using Modified Divided Differences." Lectures notes in Mathematics 362 pp22-71, Springer-Verlag (1973).

....of suitable orders and stepsizes is non trivial and has been the subject of much research. Given suitable methods to compute and advance the integration formulae, it can make the difference between a good and a bad code. A number of different approaches have been suggested for this problem [1, 2, 4, 5, 8, 9, 10, 11]. This work concentrates on and extends two of these: one implemented by Shampine in his code RDEAM [11] and a second, simpler, version based on the difference between a predictor and corrector formula. Communicated by Prof. C. T. H. Baker, this document originally appeared as an IWR preprint ....

.... following error estimator h (g k 1;1 Gamma g k;1 )OE k 1 (n 1) 1) presents itself naturally where h = t n 1 Gamma t n is the current stepsize, g k 1;1 and g k;1 are two integration coefficients computed by a set of recurrence relations , and OE k 1 (n 1) a modified divided difference [8] for the step [t n ; t n 1 ] incorporating the predictor value f n 1 = f(t n 1 ; y n 1 ) This formula is used as the basis of error control. In particular, to obtain a suitable value for h, assuming that the solution s derivatives do not vary wildly between integration steps, a variant [11] ....

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Krogh, F. T. "Changing stepsize in the integration of differential equations using modified divided differences." Lectures notes in Mathematics 362 pp22-71, Springer-Verlag (1973).

....to the initial singularity, a; Q a ) 0; 0) is not suitable 3 as a starting point. Alternative starting values (i.e. initial conditions) should be taken either from the equation (3) or from the integration coefficients (typically obtained by recurrence relations) computed by the host ODE code [10, 12] to advance the solution. In this latter case, such choices could be based on the stepsize from the last step or a value for the projected step obtained by a standard stepsize estimator. Since stepsize control requires typically only a very crude approximation to h , it could be obtained using ....

....reasons: ffl they provide a simple numerical approach to an otherwise difficult polynomial rootfinding problem and ffl their accuracy can be tailored to the accuracy required by the stepsize strategy. Expensive full precision evaluations of Q(h) using recurrence relations of the type outlined in [10, 12] are thus avoided. The direct solution of (4) in the Adams EPS case is, however, complicated by the singularity at h = 0. As we now show, this does not occur for EPUS schemes: they are singularity free. Theoretically, it is hoped that equations of the form (4) may also provide insight into how new ....

Krogh, F. T. "Changing Stepsize in the Integration of Differential Equations using Modified Divided Differences." Lectures notes in Mathematics 362 pp22-71, Springer-Verlag (1973).

....of suitable orders and stepsizes is non trivial and has been the subject of much research. Given suitable methods to compute and advance the integration formulae, it can make the difference between a good and a bad code. A number of different approaches have been suggested for this problem [1, 2, 4, 5, 8, 9, 10, 11]. This work concentrates on and extends two of these: one implemented by Shampine in his code RDEAM [11] and a second, simpler, version based on the difference between a predictor and corrector formula. Communicated by Prof. C. T. H. Baker, this document originally appeared as an IWR preprint ....

.... error estimator h (g k 1;1 Gamma g k;1 )OE p k 1 (n 1) 1) presents itself naturally where h = t n 1 Gamma t n is the current stepsize, g k 1;1 and g k;1 are two integration coefficients computed by a set of recurrence relations 2 , and OE p k 1 (n 1) a modified divided difference [8] for the step [t n ; t n 1 ] incorporating the predictor value f p n 1 = f(t n 1 ; y p n 1 ) This formula is used as the basis of error control. In particular, to obtain a suitable value for h, assuming that the solution s derivatives do not vary wildly between integration steps, a variant ....

[Article contains additional citation context not shown here]

Krogh, F. T. "Changing stepsize in the integration of differential equations using modified divided differences." Lectures notes in Mathematics 362 pp22-71, Springer-Verlag (1973).

....slight. References for the methods used in the various codes are: for DOP853 (and for the formulas used in DXRK8) Hairer et al. 1993] for RKSUITE, Brankin et al. 1991] and [Kraut 1991] these being cited with the documentation that comes with the code downloaded from netlib) and for DIVA, [Krogh 1974] and [Krogh 1994] The test program allows for interpolation to output points when it is available, or integrating to an output point and then continuing from that point. The number of output points is not large, the following table summarizes where the codes are required to give results. K = ....

Krogh, F. T. 1974. Changing stepsize in the integration of differential equations using modified divided differences. In D. J. Lipcoll, D. H. Lawrie, and A. H. Sameh Eds., Proceedings of the Conference on the Numerical Solution of Ordinary Differential Equations, Number 32 in Lecture Notes in Mathematics, pp. 22--71. Berlin: Springer Verlag.

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