The numerical solution of delay-differential equations P2

'',qci The Numerical Solution of Delay-differential Equations by David Richard Wile Department of Mathematics University of Manchester volume II of II A thesis submitted to the University of Manchester for the degree of Doctor of Philosophy in the Faculty of Science. October 1989 290 0 Contents • Volume Two - title page  289 - contents page for volume two  290 Appendicies 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  418 291 appendix A a formal proof for (1.1.3:14) 292 A formal proof for (1.1.3:14) In this appendix we give a formal proof of the result (1.1.3:14). For any given time let the order of continuity of the component y i be Z i . Our task is thus to obtain a set of lower bounds {c i : > c i } such that (1.1.3:14) holds. If y i has no discontinuity in any derivative we write c i = co. We have already discussed how for sufficiently smooth In and log} derivative discontinuities could only arise from (i) derivative discontinuities at previous solution pOints or (ii) derivative discontinuities in other solution components. Moreover we showed in equation (1.1.3:12) that the order of continuity for any given component was greater than or equal to one plus the order of any such effects. That is Ci > min{iz i ,C i } + 1  (1) isi where once again h i denotes a lower bound on the minimum order of any deriva- tive discontinuities inherited by f t from the past solution and is set to co if no such discontinuities exist. Combined with the definition of the terms {c i }, (1) now gives > min{ hi, C . ; + 1 > min{h i ,ci} + 1  (2) isi  isi which can be used to motivate the following definition =  ci} -I- 1  (3) since it is clear that any {c i } so defined will certainly suffice as lower bounds to {C i : c i < From (3) however it is also clear that not all the components {y j : iSi} coupled to y i can have an equal effect upon c i . A component y j can 293 affect y i only if its continuity bound c i lies in the minimising set fc j : jSi, c i = c j 11. Formalising this notion we define the relation jEi  jSi A {ci = c i 1}.  (4) From this we can then define an effect tree Ei =  :  U {i}  (5) where E defines the transitive closure of E. Heuristically, by an extension of the above arguments, we may think of Ei as defining the set of points whose continuity bounds can affect the continuity bound, c i , of y i . El is clearly a tree because, since jEi  c i = c j + 1, any graph defined by E can have no loops. The point  is included for completeness. In order to prove our result (1.1.3:14) we start by presenting the following proposition: PROPOSITION 1 Let l i (j) = d(j,i)-F cj where d(j,i) denotes the minimum (directed) distance from j to i along the (directed) network defined by S. Then we have (i) ci jEi  = li(i) where E denotes the transitive closure of E. Part (i) follows immediately from the definition of / i since l(i) = d(i, i) and d(i, i) = 0 for all i, l(i) must be equal c. For part (ii) however we need the following lemma.: 294 LEMMA 1 Suppose . {.50 0 is a path from b to a such that a = so, b = sk and  = 0. n — 1 then d(b, a) = k PROOF by contradiction: Suppose d(b,a) < k. Then calling our original path, {sX_ 0 , path one we construct a second path tit) r 0  a = to, b = tk,  (7) i= 0 — 1 of length k' < k which we shall call path two. Since both paths are from b to a they must converge after some point, albeit possibly at a, before reaching a. Denoting the first such point by t q that is tq  Si = ti,  = 1 q, t q + 1 s q + i .  (8) Representing this graphically, that is (12) i = so, j = t„ { r S k 1=0 295 path one b a path two figure 1 By definition, equation (3), along path two we have e t; < c ii+ , + 1 and so by induction ci9+1 < (k' — q +1) + c b  (9) whereas for path one the stronger relation E implies c,, = c si+ , + 1 and so c, ,+ , = (k — q + 1) + cb•  (10) This however leads to a contradiction for k! < k then implies Ci g+i < Cavo  (11) which is nonsense since, in the notation of (5), s q Es q + 1 whereas t q ,Et 1 + 1 . Thus k' < k cannot be true and the result then follows. [] We can now use this result to prove part (ii) of Proposition 1: PROOF of Proposition 1, part (ii). Once again we construct a path Sk+1SSkl k = 0 • 71,— 1 296 for which we will show l(s)  To show this we first establish that l i (s k +i)  k = 0 n .Ti: li(Sk+1)  1,(4-1-1).4.>  = d(s k ,i)-F csk (13) d ( s k, i ) =  csk but sk- F iEsk  c„ = c ak+ , + 1 and, by lemma 1, d(s k + i ,i) — d(s k ,i) = 1)— k = 1 so d(sk+1,i)— d(s k ,i) =  c„÷„ <#. 1 = 1  (14) which is always true. Thus l i (s k + 1 )= l i (s k ) . The full result follows by induc- tion.[] [Note that d(j,i) is always finite since jEi  jSi]. Heuristically Proposition 1 shows that inorder to find the continuity bound, for any componet i, it is sufficient merely to know a bound for any other point in the effect tree Ei. In a numerical code however the only components for which continuity bounds Ic i ) are available are those for which the bounds are constrained by discontinuities inherited directly from lag points. Formally, that is the set of components = : c i = h i + 1, h i < ool.  (15) Thus if we can show that the intersection between this set H and the set Ei is non-empty then we can, by proposition one, write down an expression for c i in terms of known information: C 1 1( j ) vj E 7-/ n Ei.  (16) We show this by means of the following proposition, which will be the basis of our result. (18) id(j,i) lSkfk_O a = SO, b = Skim 297 PROPOSITION 2 PROOF: This result follows from the properties of Ei. As we have already observed El is a tree and so can contain no loops. Furthermore since there are only a finite number of nodes El must contain at least one terminator t, that is a point t : t i such that /Br : rEi. By the definition of E this implies cg = h t + 1 and so t E E. Thus I/ n Ei is non-empty. [1 Together, Propositions 1 and 2 imply that min{/ 2 (j)} is bounded above by c i for all j E H. To prove equality it is sufficient to prove the converse relation. We do this by the following lemma. LEMMA 2 Ci < min {li(j)} vi PROOF: Recall ci = min {l i (j)} = min {d(j,i)d- c i }.  (17) vi  v.; For any j consider the shortest distance d(j,i) from j to i. If j and i are path connected, that is if j g i, then we can construct a shortest path of length d(j,i) s k + i Ss k , k = 0 d(j,i) — 1 between them. By definition, equation (3), we then have Csk+i 5 c„ 1 (since 3 k + 1 Ss k ) which gives by induction c„ < c 30 r and so c i d(j,i)-F c i for r = d(j,i). This inequality will hold for any j and so the result follows. [J 298 The final result now follows immediately. Observing min{/1(j)} < 'ii_ vj  iEHnfi we have Ci < min 1 1 (j) < min l i (j) = c i  (19) V.7  j€HnEi and so c i = min l i (j) = min {d(j,i)+  (20) jent": Moreover since H contains H fl El and is contained within the wider set H = : h i < co} [see (1.1.3:14)], we may also write Ci = min 1 1 (j) = min {d(j,i)+ c i }.  (21) jEH  jEH which is the required result (1.1.3:14). [...]... in [17] The identifier INTRP is reserved for another routine NB' this routine has however been slightly modified for use within DELSOL These alterations are clearly marked in the routines' source listing They are (i) a revision to the initial stepsize selection strategy and (ii) the inclusion of an additional reverse-communication re-entry point (IREVCM = 9) to simplify the implementation of the interpolation... however, they also can be used to give 1 this approach can easily be extended to more general conditions 302 a measure of the effect of spurious derivative discontinuities on schemes which ignore them To demonstrate, this observe that since the Adams-Bashforth method is explicit, it will give the same end-point result when applied to either of (2) (t) F(t,y(t)) (t) G(t, g(t)) or (5) Thus, since the answer... Thus the order of the effect of the spurious discontinuity at t = is one less than the local truncation error as required Predictor-Corrector Schemes Although details are not here considered, the above analysis can readily be extended to predictor-corrector methods Using a corrector or post-corrector interpolant (c.f 2.2.2) then although the coefficients co and cl are no longer necessarily zero, the. .. affect the order of the integration, since their effects are one order less than the local truncation error A discussion of Adams-Bashforth-Moulton PECE schemes is given at the end of this appendix For simplicity constant stepsize and order are assumed Adams-Bashforth schemes Consider a (p -I- 1) step Adams-Bashforth extended-ODE method with the predictor-interpolant = — see (2.2.2) — applied to the restricted... polynomial approximation In the context of numerical differentiation RALSTON [59] shows that for any (r — 1) degree polynomial p r_ i interpolating f at t ti , 12 , t,., the O h derivative error is 1.! E i=0 where w(t) = f di 717W(t)' z! (r+k—i)(77k_i) (t) j (r k — i)! r_ 1( t-ti ) and the 171i) lie in the span of Here tv (i) (t) = By repeated applications of Leibnitz's rule [41] we can then show that 0 0 (t)... give a listing of the code DELSOL described in chapter three together with the modified version of the ODE code RDEAM which it calls - DO2QFQ 1 - and a simplified driver routine For convenience therefore this appendix is divided into three parts: Cl - a listing of DELSOL C2 - a listing of DO2QFQ1 C3 - a simplified driver for DELSOL together with example A note on variable and subroutine names Although... is 17n+1 , the corresponding error incurred by neglecting the discontinuity at is simply t 1.4_ 1 ci E— i=0 We now show that this term is 0(h 71+2 ) , (6) 21 one order less than that of the local truncation error To do this it is useful to consider the back-approximant Vi(t) in terms of integrating interpolatory polynomials See again (2.2.2) Any discrepancies in (VI) (l) on either side of t = t, simply... implementation of the interpolation formulae and the multi-stepsize algorithm decribed in (2.4) Both of these modifications are discussed in section (3.2) It should be stressed therefore that the routine DO2QFQ referred to by DELSOL and listed in C3 may differ from any NAG routine of the same name 306 0 Important note on type definitions For technical reasons connected with the word-length on IBM1 machines although... (7) still holds Applying the same procedure to the corrector stage as above, and observing that an 0(h P+2 ) perturbation in AP + , has an order hP+3 effect, it can be shown, once again that the order of the 2 spurious discontinuity at t = e is not significant 304 appendix C DELSOL source listings 305 Appendix C - DELSOL : program listing In this appendix we give a listing of the code DELSOL described...299 appendix B the effect of spurious derivative discontinuities 300 The effect of spurious derivative discontinuities In section (2.1.3) we showed how spurious derivative discontinuities could arise in DDE methods purely from the form of back approximation scheme In particular we showed in section (2.2.2) how these arose in Adams PECE methods We now, by first considering . 67 3 42 CKJUMP 56 127 3 72 CKOPT 20 73 345 CLOSE 67 52 334 DDESTP 25 7 75 346 DELSOL 135 1 309 EXPAND 42 165 391 FINDF 51 58 337 FINDLG 111 106 361 FINDTP 88 131 374 FINDZ 40 103 360 ICLEAR 7 177 397 INTRP 58 149 383 INTRP2 40 1 52 384 LAST 39 1 72 394 -LCLEAR S 179 390 LINK 12 45 331 LOCATE 57 123 370 MESO 25 5 29 323 NEGONE 7 180 398 VERLAG 48 118 367 NEWTP 24 138 377 NEXT 24 170 393 PREV 23 116 366 PSTATS 25 175 396 QJUMP 16 154 385 RCLEAR a 178 397 RLINK 23 17 317 SETCTY 1 02 61 339 SETJMP 77 68 3 42 SHRINK 22 168 3 92 SORT 30 56 336 UPDATE 17 163 390 YLINK 16 49 333 ZLINK 32 50 333 (1) (2) (3) Here. 390 LINK 12 45 331 LOCATE 57 123 370 MESO 25 5 29 323 NEGONE 7 180 398 VERLAG 48 118 367 NEWTP 24 138 377 NEXT 24 170 393 PREV 23 116 366 PSTATS 25 175 396 QJUMP 16 154 385 RCLEAR a 178 397 RLINK 23 17 317 SETCTY 1 02 61 339 SETJMP 77 68 3 42 SHRINK 22 168 3 92 SORT 30 56 336 UPDATE 17 163 390 YLINK 16 49 333 ZLINK 32 50 333 (1) (2) (3) Here the numbers (2) correspond to. in page name statements listing number ADD 73 158 387 APPROX 128 140 378 AUXILO 111 95 356 AUXIL1 43 100 358 11 67 3 42 CKJUMP 56 127 3 72 CKOPT 20 73 345 CLOSE 67 52 334 DDESTP 25 7 75 346 DELSOL 135 1 309 EXPAND 42 165 391 FINDF 51 58 337 FINDLG 111 106 361 FINDTP 88 131 374 FINDZ 40 103 360 ICLEAR 7 177 397 INTRP 58 149 383 INTRP2 40 1 52 384 LAST 39 1 72 394 -LCLEAR S 179