L. F. Shampine and P. Bogacki. The effect of changing the stepsize in the linear multistep codes. SIAM J. Sci. Stat. Comput., 10:1010--1023, September 1989.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....the form y n k ff k;i y n Gammai = h n fi k f(y n ) 2) where h n is the stepsize, k is the order and the coefficients ff k;i depend on k only. In practice codes for integrating stiff IVPs vary the stepsize h n and or order k resulting in variable step variable order BDF implementations [8], 1] At each integration step t n we must solve the nonlinear equation F (y n ) j y n OE n Gamma h n fi k f(y n ) 0; 3) where OE n = P k ff k;i y n Gammai is a known value. To solve for y n most codes use the Newton iteration or its variants in the form, n n = GammaF (y n ....

L. F. Shampine and P. Bogacki. The effect of changing the stepsize in the linear multistep codes. SIAM J. Sci. Stat. Comput., 10:1010--1023, September 1989.

....directly a problem, namely the solution of a class of non linear equation [7, 12] often considered intractable. Standard codes usually make simplifying assumptions about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11]. This report (with erratum) appeared as pre print 93 47 at the IWR, Universitat Heidelberg, Germany. Typically the so called constant stepsize assumption [11, 12] Applications of the new stepsize estimators include LMM start up strategies and stepsize recovery after discontinuities. For ....

....about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11] This report (with erratum) appeared as pre print 93 47 at the IWR, Universitat Heidelberg, Germany. Typically the so called constant stepsize assumption [11, 12]. Applications of the new stepsize estimators include LMM start up strategies and stepsize recovery after discontinuities. For large systems a relatively small amount of work in selecting a good stepsize may save expensive derivative function evaluations. The particular interest of the author lies ....

Shampine, L. F. and Bogacki, P. "The Effect of Changing the stepsize in Linear Multistep Codes." SIAM J. Sci. Stat. Comp. 10 3 pp1010-1023 (1989).

....on the true errors will be as desired. The simple stepsize error relation in Equation (6.2) will e.g. not reflect the fact that a stepsize change in step r 2 will affect the local residual in step r for the fourth order VC AM. This defect has also been investigated by Shampine Bogacki [46]. They observed that, when considering the stepsize error relation in (6.2) the effects of stepsize changes can be far from what is expected. The stepsize error relation in Equation (6.2) will not be close to the true ditto, if the stepsize is such that we are close to the numerical stability ....

....The two first items certainly involve statements that are of a subjective nature, viz. sufficiently well and good enough . The third item is, at least in our problem difficult to veri, since the true system involves the unknown quantity (v ) However, as was observed by Shampine Bogacki [46], the conventional stepsize error model (6.2) is in many cases a poor description of the true system . In our case we have a clear picture of how we would like the stepsize error model for the error estimate On to be, viz. the same as the stepsize error model for the local residual . For the ....

L. E Shampine and P. Bogacki. The effect of changing the stepsize in linear multistep codes. SIAMJ. Sci. Star. Comput., 10(5):1010-1023, September 1989.

....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 ....

....in a wide range of numerical problems. In certain cases, however, they may be unfeasible, such as during the initial or startup phase see section 11 or in response to difficult situations such as externally imposed stepsize or order changes . For an introduction to this problem, see [10]. For this reason, a number of alternative approaches to the problem of stepsize control have been proposed. 5 Solving for Although several other workers have attempted to obtain improved estimators based upon the equations (2) and (4) such an approach is inherently difficult. By retaining ....

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Shampine, L. F. and Bogacki, P. "The effect of changing the stepsize in linear multistep codes." SIAM J. Sci. Stat. Comp. 10 3 pp10101023 (1989).

....directly to the correct error coefficients. Basing on the estimate err(k) it is decided if a step is accepted or rejected. Independently of this decision, a new stepsize and a new order must be proposed for being used within the next step. The stepsize proposal bases on the one step assumption [38] le = f(t l )h k 1 l O(h k 2 l ) 29) with a slowly varying f. This assumption is far from being true if the stepsize varies too much [38] Especially, the order of the method may be reduced drastically. In many codes it is therefore tried to use sequences of constant stepsize as long as ....

....decision, a new stepsize and a new order must be proposed for being used within the next step. The stepsize proposal bases on the one step assumption [38] le = f(t l )h k 1 l O(h k 2 l ) 29) with a slowly varying f. This assumption is far from being true if the stepsize varies too much [38]. Especially, the order of the method may be reduced drastically. In many codes it is therefore tried to use sequences of constant stepsize as long as possible before an attempt is made to change the order. If all else fails, the order is reduced to 1 such that effectively a one step method is ....

[Article contains additional citation context not shown here]

L.F. Shampine and P. Bogacki. The effect of changing the stepsize in linear multistep codes. SIAM J. Sci. Stat. Comput., 10(5):1010--1023, 1989.

....directly a problem, namely the solution of a class of non linear equation [7, 12] often considered intractable. Standard codes usually make simplifying assumptions 1 about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11]. This report (with erratum) appeared as pre print 93 47 at the IWR, Universitat Heidelberg, Germany. 1 Typically the so called constant stepsize assumption [11, 12] 1 Applications of the new stepsize estimators include LMM start up strategies and stepsize recovery after discontinuities. ....

....about the previous stepsizes which may fail, resulting in step rejections and possible restarts, if these conditions are not met [11] This report (with erratum) appeared as pre print 93 47 at the IWR, Universitat Heidelberg, Germany. 1 Typically the so called constant stepsize assumption [11, 12]. 1 Applications of the new stepsize estimators include LMM start up strategies and stepsize recovery after discontinuities. For large systems a relatively small amount of work in selecting a good stepsize may save expensive derivative function evaluations. The particular interest of the author ....

Shampine, L. F. and Bogacki, P. "The Effect of Changing the stepsize in Linear Multistep Codes." SIAM J. Sci. Stat. Comp. 10 3 pp1010-1023 (1989).

....form y n k X i=1 ff k;i y n Gammai = h n fi k f(y n ) 2) where h n is the stepsize, k is the order and the coefficients ff k;i depend on k only. In practice codes for integrating stiff IVPs vary the stepsize h n and or order k resulting in variable step variable order BDF implementations [8], 1] At each integration step t n we must solve the nonlinear equation F (y n ) j y n OE n Gamma h n fi k f(y n ) 0; 3) where OE n = P k i=1 ff k;i y n Gammai is a known value. To solve for y n most codes use the Newton iteration or its variants in the form, W (l) n (l) n = ....

L. F. Shampine and P. Bogacki. The effect of changing the stepsize in the linear multistep codes. SIAM J. Sci. Stat. Comput., 10:1010--1023, September 1989.

....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 ....

....in a wide range of numerical problems. In certain cases, however, they may be unfeasible, such as during the initial or startup phase see section 11 or in response to difficult situations such as externally imposed stepsize or order changes 6 . For an introduction to this problem, see [10]. For this reason, a number of alternative approaches to the problem of stepsize control have been proposed. 5 Solving for h Although several other workers have attempted to obtain improved estimators for h based upon the equations (2) and (4) such an approach is inherently difficult. By ....

[Article contains additional citation context not shown here]

Shampine, L. F. and Bogacki, P. "The effect of changing the stepsize in linear multistep codes." SIAM J. Sci. Stat. Comp. 10 3 pp10101023 (1989).

....methods [16, 58] and spline approximations [14, 59] as well as implicit RK methods [17, 26] have also been investigated. There are several possible choices of dense output for RK methods: 1. continuous extensions [73] 2. natural continuous extensions (NCEs) 94] 7 See Shampine Bogacki [79] for example. 3. one step Hermite interpolants [78] 4. highly continuous extensions (HCEs) 47] 5. multistep Hermite interpolants [71] and 6. multistep continuous extensions [50] Higham [47] concluded that HCEs are really only appropriate for sufficiently smooth ODE problems, which would ....

L.F. Shampine and P. Bogacki, The effect of changing the stepsize in linear multistep codes, SIAM J. Sci. Stat. Comp. 10 (1989) 1010--1023.

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