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Convolution Quadrature and Discretized Operational Calculus. I Numer. Math. 52, 129-145 (1988) Numerische Mathematik Springer-Verlag 1988 Convolution Quadrature and Discretized Operational Calculus. I. C. Lubich Institut ffir Mathematik und Geometrie,Universit/it Innsbruck, Technikerstrasse 13, A-6020 Innsbruck, Austria Summary. Numerical methods are derived for problems in integral equations (Volterra, Wiener-Hopf equations) and numerical integration (singular inte- grands, multiple time-scale convolution). The basic tool of this theory is the numerical approximation of convolution integrals x f*g(x)= S f(x--t)g(t)dt (x>O) 0 by convolution quadrature rules. Here approximations to f*g(x) on the grid x=0, h, 2h ..... Nh are obtained from a discrete convolution with the values of g on the same grid. The quadrature weights are determined with the help of the Laplace transform of f and a linear multistep method. It is proved that the convolution quadrature method is convergent of the order of the underlying multistep method. Subject Classifications: AMS(MOS): 65D30, 65R20, 65L05, 44A55; CR: G1.9. 1. Introduction 1.1. Derivation of the Methods We begin our investigations with the numerical approximation of convolution integrals x f(t)g(x-t)dt (x>__O). (1.1) 0 We shall obtain methods of discrete convolution form ~, eo;(h)g(x-.jh) (1.2) O<=jh<=x 130 C. Lubich where h > 0 is the stepsize, and the quadrature weights coi(h) are the coefficients of the power series F(b(~)/h)= ~ ooj(h)~J. (1.3) j=o Here F is the Laplace transform of f and 6(~)=~bj~ J is the quotient of the 0 generating polynomials of a linear multistep method. As will be pointed out in Part II of this work, quadrature methods of this type are particularly suited to the approximation of convolution integrals whose kernel f(t) is singular or has components at different time-scales; to the numeri- cal evaluation of the expressions arising in classical operational calculus where the Laplace transform F of the convolution kernel (and not the kernel f itself) is known a priori; and to obtain stable discretizations of integral equations. From the control engineer's viewpoint our results can be seen as results on the discrete-time simulation of a continuous-time linear system given via its transfer function F(s). To explain this approach we start from the Laplace inversion formula f(t)=~n / r~ F(2)e~td2 (t>0) (1.4) which holds if, for example, we assume 7~ F(s) is analytic in a sector [arg(s- c)[ < n -- ~o with ~o < ~, c elR and satisfies there ]F(s)] <m-Is]-" for some M< oo, #>0. (1.5) The contour F can then be chosen as running from oo-e -i("-~) to oo-e i~-~) within the sector of analyticity of F(s). A condition equivalent to (1.5) is that f(t) is analytic and of (at most) exponential growth in a sector containing the positive real half-line and satisfies f(t)= O(t ~- 1) as t--, 0 within the sector. The assumption/~ > 0 guarantees in particular that f(t) is locally integrable. Examples include fractional powers, logarithms and most special functions. The approach (1.2), (1.3) now comes about in the following way: Inserting (1.4) into (1.1) and reversing the order of integration gives x 1 x S f(t)g(x-t)dt=~n i ~r F(2)~o eZtg(x--t)dtd2" (1.6) 0 We now approximate the inner integral by applying a linear multistep method to the differential equation y' = 2y + g, y(0) = 0, k k Z aJYn+J-k-=h Z flJ(2Yn+J k+g((n+j-k)h)) (n>O), (1.7) j=O j=O Convolution Quadrature and Discretized Operational Calculus. I. 131 with starting values Y-k ..... y_~=0, and with g~C[0, oo) extended by 0 to the negative real axis. Multiplying (1.7) by ~" and summing over n from 0 to oe we obtain (% (k +... + ~k)"Y(O : (flO (k +... + ilk)" (h 2' y(() + h-g(O), co co with the generating (formal) power series Y(0 = ~Y, (", g(0 = ~g(n h) (". Solving 0 0 this equation for Y(0 we find that y, is the n-th coefficient of the power series (b(O/h- 2) -1 g((), where 6 (0 = (~o ~k +... + C~k _ i ~ + C~k)/(flO~k § + fig - I ~ + fig)" (t .8) Hence the resulting approximation of (1.6) at x = nh is the n-th coefficient of 1 /6(0 2)-lg(Od2=F(6(~h) ) (1.9) where the equality holds by Cauchy's integral formula. The coefficients of the right-hand side of (1.9) are the Cauchy product of the two sequences {~o;(h)} of (1.3) and {g(jh)}. This gives finally (1.2). The above derivation indicates also how to obtain error estimates: One studies first the error introduced by approximating the inner integral in (1.6) by the multistep method (1.7), then multiplies the obtained error bound by the bound (1.5) of F(2) and integrates along the contour F. This will actually be carried out in Sects. 2 and 3. Of the linear multistep method we shall assume that it is A(a)-stable with a>(p of (1.5), stable in a neighbourhood of infinity, strongly zero-stable and consistent of order p. In terms of 6(0, these conditions can be expressed as: 3(0 is analytic and without zeros in a neighbourhood of the closed unit disc ]~1__<1, with the exception of a zero at ~ = 1. (1.10a) [arg6(0l<n-c~ for [~l<l, forsome c~>q~. (1.10b) 1 76(e-h)=l+O(h p) forsome p>l. (1.10c) n Well-known examples are the backward differentiation formulas of order p < 6, given by 3(~)= ~-(1--0 i, with e = 90 ~ 90 ~ 88 ~ 73 ~ 51 ~ 18 ~ for p--1, ..., 6, respectively, i= 1 Condition (1.5) on the transfer function (or symbol) F(s) of the convolution (1.1), and condition (I.10) on the discretization method will serve as the main assumption for all results of this paper. Remark. (1.3) is well-defined if 5o/h is in the domain of analyticity of F(s). Since 30 >0 by (1.10), this is satisfied at least for sufficiently small h >0. The integrals in (1.6) and (1.9) are absolutely convergent, because ~ > 0 in (1.5). 132 C. Lubich 1.2. Symbolic Notation and Historical Remarks It will be convenient to use transfer function notation for the convolution (1.1), x F(s)g(x)=f*g(x)= ~ f(t)g(x-t)dt (s: complex variable). (1.i1) 0 We use the abbreviation Sh=(~(~)/h and employ discrete transfer function nota- tion for (1.2), (1.3), F(Sh)g(x)= ~, coi(h)g(x--jh). (1.12) O<jh<x Note that F(Sh)=~'c%(h)( ~ is the discrete Laplace transform of the sequence 0 {coj(h)}.) With this notation we obtain from (1.9) the "Cauchy integral formula" 1 F (s,) g (x) = T~ ~ F (2~). (sh - .~) -1 g (x). d ,~, (t. 13 ) and the same relation, with s formally in place of sh, holds by (1.6), since (s- 2)- is the Laplace transform of e ~'. We shall use the notation (1.11) also for distributional convolution, in partic- ular for differentiation sg=~'.g=g'+g(0).~ (Dirac's 6). (1.14) As usual, we identify distributions with functions outside their singular support (which wilt be at most {0} in our applications), so that (sg)(x)=g'(x) for x>0. The derivative (1.14) is discretized by the backward difference approximation 1 Shg(X)=~ Z 6~g(x--jh). (1.15) O<=jh <=x With this in mind, the discretization process leading from (t.11) to (1.12) has a simple interpretation: The symbol of differentiation, s, is replaced by the symbol of a backward difference quotient, Sh. This idea of a discretized operational calcu- lus has actually very old origins and seems to have been rediscovered several times. Of particular historical interest in this context is the work of Post [12] who has shown pointwise convergence of F(Sh) g(x) as h~0 for the simple difference quotient sh g(x)=(g(x)-g(x--h))/h, thereby generalizing earlier work of Liouville [10] and Gr/inwald [8] on fractional integrals, i.e. the case F(s)= s -u. Post's work appears to have gone unnoticed in numerical analysis (in which he himself showed no interest), and the author of the present paper is not aware of any results in the literature going beyond those of Post. Convolution Quadrature and Discretized Operational Calculus. I. 133 1.3. Outline of the Paper In Sect. 2 we study the convergence properties of the linear multistep method (1.7) (i.e., of (Sh--2)- 1 g(x)) for 2 varying in a sector of the complex left half-plane. Such problems have previously been studied in the numerical analysis of para- bolic differential equations by Ztfimal [15] and Crouzeix and Raviart [2], see also Gekeler [7]. Here we give (and require) stronger estimates in terms of the input g, using different techniques. Section 3 contains the first main results. The convergence properties of con- volution quadrature rules F(Sh)g(x) are now immediately obtained by integra- ting the error bounds of Sect. 2 along the contour F. The convergence order of the underlying multistep method, p, is obtained for geCP[0, oo) by adding a few correction terms to F(Sh) g(x), p-2 F(Sh)~g(x)=F(sh)g(x)+ ~ w,j(h)g(jh) at x=nh.
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