D. R. Will'e and C.T.H. Baker, DELSOL - a numerical code for the solution of systems of delay-differential equations, Appl. Numer. Math. 9 (1992) 223--234.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....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 in solving delay differential equations [14]. 2 Adams formulae 2.1 Predictor corrector schemes Given an ODE y 0 (t) f(t; y(t) the k th order Adams Bashforth and (k 1) th order Adams Moulton methods to advance a numerical solution fe y i y(t i )g across a step [t n ; t n 1 ] may be written as e y n 1 : e y n Z t n 1 t n P k;n ....

Will'e, D. R. and Baker, C. T. H. "DELSOL -- a numerical code for the solution of systems of delay-differential equations." App. Nummer. Math. 9 pp223-234 (1992). lmm.tex (2.4), 94/03/23 10:52:37 12

....codes were developed by modellers or numerical analysts. The second period can be characterized by the availability of more sophisticated DDE solvers. The Numerical Algorithms Group (Oxford) supported, in part, the construction of the codes written by Paul (Archi) 149] and Will e (DELSOL) [179]. The major problems that the designers of such codes try to accommodate are: automatic location or tracking of the discontinuities in the solution or its derivatives, efficient handling of any stiffness (if possible) dense output requirements, control strategy for the local and global error ....

.... is (see Table 1) based on the successful Dormand Prince fifth order RK method for ODEs due to Shampine [170] and a fifth order Hermite interpolant [146] In addition to Archi [149] which is available from the internet, we mention DDESTRIDE (Baker, Butcher Paul [15] DELSOL (Will e and Baker [179]) DRKLAG6 (Corwin, Sarafyan Thomson [42] SNDDELM (by Jackiewicz Lo [97] and the code of Enright and Hayashi [62] Other approaches may be found in the literature. Fourth order RK methods and two point Hermite type interpolation polynomials were used by Neves [140] and algorithms based on ....

[Article contains additional citation context not shown here]

D. R. Will'e and C.T.H. Baker, DELSOL - a numerical code for the solution of systems of delay-differential equations, Appl. Numer. Math. 9 (1992) 223--234.

....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 in solving delay differential equations [14]. 2 Adams formulae 2.1 Predictor corrector schemes Given an ODE the k th order Adams Bashforth and (k 1) th order Adams Moulton methods to advance a numerical solution fe y i y(t i )g across a step [t n ; t n 1 ] may be written as P k;n (t) dt P k 1;n 1 (t) dt respectively, where P i;j ....

Will'e, D. R. and Baker, C. T. H. "DELSOL -- a numerical code for the solution of systems of delay-differential equations." App. Nummer. Math. 9 pp223-234 (1992). lmm.tex (2.4), 94/03/23 10:52:37 12

....a general problem. Immediately following derivative discontinuities, intervals containing no discontinuities may be hard to guarantee. The lag point ff(t) may immediately recross j without being detected. Numerically, codes typically ignore such cases. An example of this occurs in DELSOL [10, 13]. Where full lag function derivative information is available, however, one possible approach to this problem makes use of Roll e s theorem [5] Such techniques have also been proposed for ODE g stop methods in [1] Providing ff(t) is continuous, if a second root ff( j exists, ff (j) ....

Will'e, D. R. and Baker, C. T. H. "DELSOL -- a numerical code for the solution of systems of delay-differential equations." App. Numer. Math. 9 pp223-234 (1992).

....is thus given Sequences of constant stepsizes also make way for more efficient ways to evaluate the integration coefficients used to advance the integration and simplify order control. The author s interest in the study of such cases arises in the solution of delaydifferential equations. See [12, 14]. i= Gamma1 (t Gamma t n Gammai ) dt: 5) t n 1 ; t n ; t n Gammak 1 ] here denotes the (k 1) st Newton divided difference through the points f(t i ; y i ) t n 1 ; y n 1 ) i = n; n Gamma k 1g. The required stepsize estimation thus reduces to a problem of the form ....

Will'e, D. R. and Baker, C. T. H. "DELSOL --- a numerical code for the solution of systems of delay-differential equations." App. Num. Maths. 9 p223-234 (1992.

....a frequent source of such problems is population modelling that leads to the Volterra Lotka equations, or generalizations thereof. Our results extend those for systems of delay differential equations (DDEs) that are well known (see [5, 11] and have been utilised in the design of some codes [8, 9]. The Department of Mathematics. The University of Manchester, Manchester M13 9PL y Now at Novartis Services AG, Basle, Switzerland z The first author acknowledges financial support from EPSRC under grant GR L 35218. The authors thank Dr. C.A.H. Paul for his comments on the paper, which was ....

Will'e, D.R., and Baker, C.T.H., DELSOL -- A numerical code for the solution of systems of delay differential equations. Appl. Num. Math. 9 (1992), 223--234.

....Features of Archi code. by modellers or numerical analysts. The second period can be characterized by the availability of more sophisticated DDE solvers. The Numerical Algorithms Group (Oxford) supported, in part, the construction of the codes written by Paul (Archi) 149] and Will e (DELSOL) [179]. The major problems that the designers of such codes try to accommodate are: automatic location or tracking of the discontinuities in the solution or its derivatives, efficient handling of any stiffness (if possible) dense output requirements, control strategy for the local and global error ....

.... is (see Table 1) based on the successful Dormand Prince fifth order RK method for ODEs due to Shampine [170] and a fifth order Hermite interpolant [146] In addition to Archi [149] which is available from the internet, we mention DDESTRIDE (Baker, Butcher Paul [15] DELSOL (Will e and Baker [179]) DRKLAG6 (Corwin, Sarafyan Thomson [42] SNDDELM (by Jackiewicz Lo [97] and the code of Enright and Hayashi [62] Other approaches may be found in the literature. Fourth order RK methods and two point Hermitetype interpolation polynomials were used by Neves [140] and algorithms based on ....

[Article contains additional citation context not shown here]

D. R. Will'e and C.T.H. Baker, DELSOL - a numerical code for the solution of systems of delay-differential equations, Appl. Numer. Math. 9 (1992) 223--234.

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D.R. Will'e & C.T.H. Baker, DELSOL -- A numerical code for the solution of systems of delay differential equations. Appl. Numer. Math., 9 (1992), pp. 223--234.

....the smoothness of the solution of a VFDE having a lagging argument. The solution y is smooth on each interval (oe r ; oe r 1 ) r 0. This observation can be extended to systems of DDEs with multiple lags, and a network dependency graph can be constructed to see how discontinuities propagate (see [174, 175]) 4.2 Dynamic behaviour There is a growing literature on oscillation [97] bifurcation, and chaos [73] in DDEs, and on the stability [140] of certain types of solution. The numerical approximations should emulate important features of the analytical solutions. In the literature, there is ....

....depends upon the degree of smoothness of the solution in a neighbourhood of oe . There will be instances where one has to cross a point of discontinuity, and what is required is that in doing so the error incurred remains small and is not subsequently magnified to an unacceptable degree. See [172, 174, 175], etc. 7.4 Error control and practical codes Error control mechanisms used in the numerical solution of ODEs are sometimes adapted to methods for DDEs or VIDEs. The main strategies are based on: ffi comparison formulae to estimate truncation error (for example, using embedded formulae) ffi ....

Will'e, D.R. & Baker, C.T.H. (1992). DELSOL -- A numerical code for the solution of systems of delay differential equations. Appl. Num. Math., 9, 223--234.

....a general problem. Immediately following derivative discontinuities, intervals containing no discontinuities may be hard to guarantee. The lag point ff(t) may immediately recross 0 j without being detected. Numerically, codes typically ignore such cases. An example of this occurs in DELSOL [10, 13]. Where full lag function derivative information is available, however, one possible approach to this problem makes use of Roll e s theorem [5] Such techniques have also been proposed for ODE g stop methods in [1] Providing ff(t) is continuous, if a second root ff( j exists, ff 0 ....

Will'e, D. R. and Baker, C. T. H. "DELSOL -- a numerical code for the solution of systems of delay-differential equations." App. Numer. Math. 9 pp223-234 (1992).

.... n ) h n X i=1 b i ( f(t n c i h n ; e y(t n c i h n ) e y(ff(t n c i h n ; e y(t n c i h n ) Natural formulae for dense output have an aesthetic appeal (see Hill [48] and have motivated several DDE codes, such as those written by Bader 6 [6] Bock Schloder [22] and Will e [91] based on LMF methods, and those of Baker, Butcher Paul [13] Hairer, Wanner N rsett [44] Neves [68] Neves Thompson [69] and Paul [75] based on RK methods. An Adams code of Lo Jackiewicz for NDEs appears in [65] p.377 et seq. 3.2 Stability and Order It is important to stress that: 1. ....

....for ODEs involves assessment of stability and accuracy, but has also been supplemented by considerable practical experience, experimentation and heuristics. The same considerations apply in the construction of DDE codes. However, in the case of variable order codes such as DELSOL (an LMF code) [91] and STRIDE (a RK code) 13, 25, 26] the effect on the error in the solution of a (significantly) lower order dense output for the delayed term than the current order of the method is often ignored; in practice, the accuracy of the method is more important than order. The same issues arise for ....

[Article contains additional citation context not shown here]

D.R. Will'e and C.T.H. Baker, DELSOL -- a numerical code for the solution of systems of delay differential equations, App. Num. Math. 9 (1992) 223--234.

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