Grossmann, Ch.

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Institute of Mathematics AS CR, v. v. i.

EQUADIFF 7Christian GrossmannQuasi-Newton updates in row-methods for avoiding JacobiansIn: Jaroslav Kurzweil (ed.): Equadiff 7, Proceedings of the 7th Czechoslovak Conferenceon Differential Equations and Their Applications held in Prague, 1989. BSB B.G.Teubner Verlagsgesellschaft, Leipzig, 1990. Teubner-Texte zur Mathematik, Bd. 118.pp. 252--255.Persistent URL: http://dml.cz/dmlcz/702331Terms of use:© BSB B.G. Teubner Verlagsgesellschaft, 1990Institute of Mathematics of the Academy of Sciences of the Czech Republic providesaccess to digitized documents strictly for personal use. Each copy of any part of thisdocument must contain these Terms of use.This paper has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The CzechDigital Mathematics Library http://project.dml.czQUASI-NEWTON UPDATES IN ROW-METHODS FOR AVOIDING JACOBIANS GROSSMANN Ch., , GDR 1. Introduction Numerical methods for solving stiff ordinary differential equations require high order of consistency as well as good stability properties. As a rule implicit methods only can treat stiff systems with acceptable step sizes. Replacing the nonlinear systems arising in implicit methods by some truncated Newton-method one obtains linearly implicit schemes. The ROW-methods form a quite arbitrary class of linearly implicit methods for solving stiff ODE's generalizing this idea. In this paper we report on some new approach jointly developed with Dr.W.Burmeister and Dr.S.Scholz which can be characterized by some truncated quasi-Newton method applied to implicit techniques. We use one quasi-Newton iteration only in each integration step of the ODE solver. Here, we sketch the main ideas only. For further details the interested reader is referred to [1], [2]. Let us consider the following initial value problem y'(x) = f(y(x)) , x€(0,X), y(0) = yo . (1) We approximate the solution of this problem by a discretization tech-nique with a fixed stepsize h>0. Steihaug/Wolfbrandt [6] proposed the following s stage W-method as a generalization of ROW-schemes: s yn +1 = yn + h 7 Wi ki , n i = l with ki , n as the solution of the linear systems i-J, i-1 (I- hV-An) ki.n = f(yn + h 2 . ^ i j kJ . n > + «A" ^ /* J kJ . n • <2) j=l j=l Here wi , * i j , f - J > if anc* A" denote real parameters and matrices, respectively, defining the method. With the choice An = f'(yn) this method forms a ROW-method as considered in [4]. The 252 1 f 2 f ' f 3 f " f f , f f f 4 f " ' f f f , f . f » » f f f " f f ' í , f > f » f » f order p of consistency of method (2) in the usual way can be derived from the the local truncation error s y(x+h) - y(x) - h 2 . wi ki(y(x)) (3) i=l with ki(.) defined analogously to (2). Expanding the functions in (3) according Taylor's formula we obtain powers of the stepsize h mul-tiplied by some linear combinations of so called elementary differen-tials (e.d.). Using (2), we obtain the classical e.d.'s known from ROW-methods as well as additional e.d.'s. The following table sum-marizes these e.d.'s up to the order p=4. p classical e.d.'s additional e.d.'s Af f Af, Af f, AAf f AAf, f'fAf, AAAf, Af Af, Af'ff AAff, f f A f , f A f f , Af'ff Here the indices n are omitted to make the notation more simple. A stands for An and f stands for f (yn) e.g. We investigate the order of consistency of the proposed method by checking up to which order all of the occurring e.d.'s are vanishing. In the case of ROW-methods all e.d.'s containing the matrices A coincides with appropriate clas-sical e.d.'s because of A=f . In the general case if the matrices A are not related to the Jacobians f, however, the additional e.d.'s can't be canceled. As a consequence there does not exist any 3-stage method of order 3 as shown in [5]. However, the quasi-Newton update reduces the number of occurring condition significantly as shown later. 2. Eliminatipn of gproe additipnal e.d.'g by guagi-Eewton updateg In quasi-Newton methods the matrices An are updated such that some secant equation holds. We denote Sn := yn - yn-1 , dn := (fn - fn-1 )/h (3) Then the equations An sn = h dn (4) 253 are used as secant equation. Furthermore the generated matrices An are supposed to be uniformly bounded for any h>0 and for arbitrary in-dices n. We guarantee (4) to be satisfied by using the idea of Broyden's updating (see [5] e.g.). This leads to formulas of the type ( h dn - Bn Sn ) qn An = Bn + = (5) qn ' Sn with arbitrary matrices Bn and vectors qn In [1], [2] we inves-tigated especially the cases qn = sn and B n = B or Bn = An-1 The later technique results in the updating formula ( h d n - A n - i sn )sn An = An-i + -. with some given starting matrix Ao. We obtain the approximation property An fn = fn fn + 0(h) as an essential consequence of the proposed construction. This property will be used later to prove the order of consistency of the new method. Next, we sketch the idea of deriving the expansion of the local trun-cation error. We restrict us to the order p up to 4 for the sake of simplicity of the representation. From the differential equation (1) and standard Taylor's expansion we obtain yn-i = yn - hfn + 1/2 h2(f'f)n - 1/6 h3(f"ff + f'f'f)n + 0(hP) and fn-l = fn - h(f'f)n + 1/2 h2(f"ff + f'f'f)n - 1/6 hMf'"fff + f'f"ff + 3f"ff'f + f'f'f'f)n + 0(hP). Using (3), now, we can find expansions for sn and dn. Finally, with (4) this leads to the representation Anfn = (f'f)n - 1/2 h(f"ff + f'f'f)n + l/2hAn(f'f)n + 1/6 h2(f'"fff + f'f"ff + 3f"ff'f + f'f'f'f)n - 1/6 h2An(f"ff +f'f'f)n + O(hP'l). This relation enables us to replace some of the additional e.d.'s oc-curring in the given table by classical ones. Thus, the number of or-der condition can be reduced significantly. 254 3 . Convergence results for the new methods We can show some efficient methods to exist by means of direct inves-tigation of the resulting order conditions. The main results derived in [1] are: Theorem 1: Let the initial matrix Ao satisfy the condition Aofo = fofo + 0(h) and the matrices An be constructed by (5) with fixed Bn . Then the proposed method (2) with 2 stages has the order P=3 if the parameters of the method are selected according f= 1/2(1+1/{T ), tCzi = 1.5773 5026, jfzi =-2.9856 4060 wi = 0.7950 1732 W2 = 0.2049 8268. For the parameters given above the related ROW-method is know to be strongly A-stable. The method considered here has the same stability behavior with respect to the linear test equation because the updating does not change the initial matrix Ao=fo and the method forms a ROW-method for this special problem. The determination of the given parameters from the consistency conditions is rather technical. We refer the interested reader to [1], [2] for further details. Furthermore in [1] a family of 4-stage methods of order p=4 has been constructed. Finally in [2] the variable stepsize case is inves-tigated. Numerical examples (see [1]) show a good coincidences of the theoreti-cal results with the computer experiments. Referenceg [1] Burmeister,W.; Grossmann,C.; Scholz,S.: On quasi-Newton updates in ROW-methods. Preprint TUD 07-13-88. [2] Burmeister,W.; Scholz,S.: On quasi-Newton updates in ROW-methods with variable stepsize. Preprint TUD 07-07-89. [3] Burrage,K.: A family of Runge-Kutta methods for solving stiff differential equations. BIT 18(1978), 22-41. [4] Kaps,P.; Rentrop,P.: Generalized Runge-Kutta methods of order four with stepsize control for stiff ODE's. Numer.Math. 33(1979), 55-68. [5] Schwetlick,H.: Numerische Loesung nichtlinearer Gleichungen. Verlag d.Wissenschaften, Berlin, 1979. [6] Steihaug,T.; Wolfbrandt,A.: An attempt to avoid Jacobian and nonlinear equations in the numerical solution of stiff differential equations. Math.Comp. 33(1979), 521-534. 255 

Quasi-Newton updates in row-methods for avoiding Jacobians

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Quasi-Newton updates in row-methods for avoiding Jacobians

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Institute of Mathematics AS CR, v. v. i.