The numerical solution of delay-differential equations P1

The Numerical Solution of Delay-differential Equations by David Richard Wille Department of Mathematics University of Manchester volume I of II A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science. October 1989 2 0 Contents • Volume One - title page  1 - contents  2 - abstract  6 - declaration and statement  8 - dedication  9 - acknowledgments  10 - aims  11 - notation  12 Chapter one - Introduction and ODE methods - foreword  16 1.1 an introduction to delay-differential equations  17 1.1.1  introduction  18 1.1.2 extensions and uniqueness  28 1.1.3  the propagation of derivative discontinuities through systems of delay-differential equations  31 1.2 ODE methods  42 1.2.1 Runge-Kutta schemes  43 1.2.2 linear multistep formulae  46 1.2.3 local error control  52 3 Chapter two - DDE methods - foreword  59 2.1 introduction, discontinuities and stepsize control  61 2.1.1  introduction  62 2.1.2 formalisation  64 2.1.3 derivative discontinuities  65 2.1.4 accuracy  74 2.2 linear multistep and predictor corrector methods  75 2.2.1  linear multistep formulae  76 2.2.2 predictor-corrector methods  79 2.3 stability  84 2.4 stepsize control and state-dependent problems  98 2.4.1 state-dependence  99 2.4.2 multistcpsize methods  103 2.4.3 a continuity requirement  112 2.5 extended ODE-techniques and the method of steps  114 2.6 an alternative scheme for error control  117 2.6.1 ordinary differential equations  118 2.6.2 systems of equations  136 2.6.3 delay-equations  137 2.7 detecting derivative discontinuities  152 4 Chapter three - DELSOL - foreword  167 3.1 design overview  169 3.1.1  introduction and foreword  170 3.1.1.1  foreword  170 3.1.1.2 program communication  171 3.1.1.3  representation  174 3.1.2 overview  180 3.1.2.1  storage  182 3.1.2.2 back-approximation and lag evaluation  188 3.1.2.3 secondary stepsize control  194 3.2 implementation  208 3.2.1 an introduction to RDEAM  209 3.2.2 interpolation formulae  214 3.2.3 stcpsize modifications  218 3.2.4 error estimation, stcpsize and order control  222 3.2.5 stcpsize selection following secondary stepsize controls  229 3.2.6 secondary order control  234 3.3 numerical results  238 3.3.1 comparative results  241 3.3.2 further examples  253 3.3.3 DELSOL - illustrations  273 3.3.4 extensions  281 5 0 Contents • Volume Two - title page  289 - contents page for volume two  290 Appendictes A  a formal proof for (1.1.3:14)  291 B  the effect of spurious derivative discontinuities  299 C  DELSOL source listings  304 Cl - subroutine DELSOL  307 C2 - subroutine DO2QFQ  399 C3 - a simplified driver for DELSOL together with example.  407 D  -  a listing of the root location algorithm discussed in (2.7)  415 Bibliography  418 6 0 Abstract Delay-differential equations (DDE's) arise in many fields of science and engineering.  In this thesis we consider the development of numerical software for the solution of such problems. Our discussion opens with a brief introduction to the theory of delay-differential equations. Attention is paid to features relevant to numerical codes. In particular a model for the propagation of derivative discontinuities through systems of equations is presented. Following a short resumd of standard techniques for the solution of ordinary differential equations (ODE's), we then consider the application of ODE software to evolutionary DDE's. Special attention is paid to the occu r rence, effect and accommodation of derivative discontinuitics and the approach is illustrated for linear multistep and predictor-corrector methods. After discussing stability, some problems specific to state-dependent delay-problems are considered before a brief comparison with the 'method of steps' as described by El'sgorts. An new alternative error control strategy for ODE and DDE schemes based upon a variational-type error analysis is then presented, followed by a discussion of the problems inherent in detecti ng derivative discontinuities. 7 We conclude by presenting a variable-order variable-step numerical routine, derived from an existing reverse communication Adams PECE ODE code, suitable for the solution of systems of delay-differential equations. A novel representation for the differential equation is used to acknowledge structural differences between delay- and ordinary differential systems.  Special attention is also paid to the organisation of lag function evaluations, back-solution approximation and order and stepsize controls. Finally we present a selection of numerical examples and a discussion of the codes application to more general delay- and to neutral-differential problems. 8 0 Declaration No portion of the work referred to in this thesis has been submitted in support of an application for another degree or qualification of this or any other university or institute of learning. o Statement Since obtaining a BSc. in Mathematics in 1985, David Wille has studied under Prof. C. T. H. Baker in the department of mathematics at the university of Manchester. In 1986 he obtained the degree of MSc. in Numerical Analysis and Computation and held the position of temporary lecturer for the academic year 1988-89 within the above department. He is currently employed as a research associate. 9 to my mum, my dad and my sister Sian 10 0 Acknowledgments I would like to start by expressing my thanks to Prof. C. T. H. Baker who's constructive comments and lively interest have been invaluable throughout the preparation of this thesis. Many thanks also to all those in Manchester and at NAG who have helped me through the course of my work. In particular I would like to thank the Numerical Algorithms Group for making available a copy of their code DO2QFQ and Julia and Lynn for their expert and rapid typing. This work was supported in part through a CASE award from the Science and Engineering Research Council in collaboration with the Numerical Algorihms Group (UK) Ltd. [...]... Introduction p Delay-differential equations In this thesis we are concerned with the numerical solution of delay-differential equations (DDE's) Delay-differential equations may best be regarded as extensions of ordinary differential equations (ODE's) in which the solution derivative y (t) is allowed to depend not only on the current solution point (t,y(t)), but also on values of the solution at previous... however, the development of general purpose robust software to be a suitable consideration for numerical analysts the Numerical Algorithms Group Plc 12 a Notation • Mathematical notation R the set of real numbers n x R the set Rn 1=1 Z the set of integers Z+ the set of positive integers C the set of complex numbers the set cn {ieZ:i  l} f x + iy : x,y e R 1 n x C i=1 C ( A 4 B ) the set of continuous... axis, and the second along the vertical 15 Chapter one introduction and ODE methods 1R Chapter one - foreword In chapter one we present some of the background theory and numerical methods necessary for the numerical treatment of delaydifferential equations We start - in (1.1.1) - with a short introduction to the theory and background of delay -equations We present an existence and uniqueness proof in section... discontinuies at the initial point or in the initial set, to understand their subsequent distribution it is sufficient to understand how they are propagated through the solution This is is the subject of section (1.1.3) Information about the solutions continuity is of crucial importance to numerical methods A failure to correctly accommodate derivative discontinuities can undermine the numerical formulae... advanced- equations are sometimes collectively referred to as 'differential equations with deviating arguments' 31 1.1.3 The propagation of derivative discontinuities in systems of delay-differential equations 0 Introduction Since the presence of derivative discontinuities can clearly affect many of the schemes used to integrate DDE's (2.1.3), for numerical methods it is often useful to know the position... above assertion These two modes of propagation (i) and (ii) will correspond to our definitions of strong and weak coupling Consider case (i) first If the i th component, y(t) of (11), is to inherit a discontinuity (directly) from some other component, the j th say, then it is clear from the above equations that yj(t) must appear and be referenced in either the argument list of fi, or of a lag function... required to equations from other sections, the equation number is then prefixed by the appropiate section number Thus (1.1.3:14) denotes equation (14) in section (1.1.3) The use of square brackets, eg [25], is reserved for references o Graphs Graphs are refe rred to in the text as cartesian products Thus t x y(t) denotes the graph of y as a function of t The first variable is always plotted along the horizontal... if (i) it inherits a discontinuity from some other component of y(t) or (ii) one of its lag points a— mij discontinuity lies on a past (t,y(t)) Writing z = y (a ) we now show this analytically Using the m ij mij m ij continuity of fi and {oci} , and the notation of (11), we by first multiple applications of the chain rule, and then by the mean value theorem obtain I (r+1) y i (ti) - () r+1 y i (t2)... dependence of the system It should be noted (of course) that two nodes are strongly dependent only if there is a direct link between them Although, for example, in figure (2) a discontinuity in y2 may affect the continuity of y4 the indices 2 and 4 are not strongly coupled as the discontinuity must cross more than one link to pass between them Thus, to test whether 35 some component, yj, can affect some other... Thus the equation )1' ( t ) = fO,Y(04(t-1)), t  0 (1) is an example of a DDE since y ' (t) can depend directly not only on the current values t and y(t) but also the delayed - term y(t-1) In general the derivative y (t) can depend on any [finite number] of past solution points, or lag points These are themselves defined by lag functions (see below) The lag points may vary in position not only with the . foreword  16 7 3 .1 design overview  16 9 3 .1. 1  introduction and foreword  17 0 3 .1. 1 .1  foreword  17 0 3 .1. 1.2 program communication  17 1 3 .1. 1.3  representation  17 4 3 .1. 2 overview  18 0 3 .1. 2 .1  storage  18 2 3 .1. 2.2. Introduction and ODE methods - foreword  16 1. 1 an introduction to delay-differential equations  17 1. 1 .1  introduction  18 1. 1.2 extensions and uniqueness  28 1. 1.3  the propagation of derivative. for error control  11 7 2.6 .1 ordinary differential equations  11 8 2.6.2 systems of equations  13 6 2.6.3 delay-equations  13 7 2.7 detecting derivative discontinuities  15 2 4 Chapter three