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Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials

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Numerical Integration of Ordinary Differential Equations Based on Trigonometric Polynomials Numerische Mathematik 3, 381 -- 397 (1961) Numerical integration of ordinary differential equations based on trigonometric polynomials By WALTER GAUTSCHI* There are many numerical methods available for the step-by-step integration of ordinary differential equations. Only few of them, however, take advantage of special properties of the solution that may be known in advance. Examples of such methods are those developed by BROCK and MURRAY [9], and by DENNIS Eg], for exponential type solutions, and a method by URABE and MISE [b~ designed for solutions in whose Taylor expansion the most significant terms are of relatively high order. The present paper is concerned with the case of periodic or oscillatory solutions where the frequency, or some suitable substitute, can be estimated in advance. Our methods will integrate exactly appropriate trigonometric poly- nomials of given order, just as classical methods integrate exactly algebraic polynomials of given degree. The resulting methods depend on a parameter, v=h~o, where h is the step length and ~o the frequency in question, and they reduce to classical methods if v-~0. Our results have also obvious applications to numerical quadrature. They will, however, not be considered in this paper. 1. Linear functionals of algebraic and trigonometric order In this section [a, b~ is a finite closed interval and C~ [a, b~ (s > 0) denotes the linear space of functions x(t) having s continuous derivatives in Fa, b~. We assume C s [a, b~ normed by s (t.tt IIxll = )2 m~x Ix~ (ttt. a=0 a~t~b A linear functional L in C ~[a, bl is said to be of algebraic order p, if (t.2) Lt': o (r : 0, 1 ..... p)' it is said to be of trigonometric order p, relative to period T, if (1.3) Lt=Lcos r 2z~ t =Lsln r ~-.t =0 (r=t,2 ..... p). Thus, a functional L is of algebraic order p if it annihilates all algebraic poly- nomials of degree =<p, and it is of trigonometric order p, relative to period T, if it annihilates all trigonometric polynomials of order =< p with period T. Functionals of trigonometric order p are comparable with those of algebraic order 2p, in the sense that both involve the same number of conditions. The * Oak Ridge National Laboratory, operated by Union Carbide Corporation for the U. S. Atomic Energy Commission, Oak Ridge, Tennessee. 382 WALTER GAUTSCtlI : relationship turns out to be much closer if we let L depend on the appropriate number of parameters. In fact, consider functionals of the form (1.4) L x=fll Ll X +... + ~2pL2p x + L2p+l x, where L~ (2 >--t) are fixed linear continuous functionals in Cs [a, bJ and //~ real parameters. Then the following theorem holds. Theorem 1. Let the ]unctionals L~ in (1.4) satis/y the /ollowing conditions: (i) L~t=0 (2=t,2 ..... 2p+t). (ii) There is a unique set o/ parameters, fla=flo, such that the/unctional L in (1.4) is o/algebraic order 2p, that is to say, (t.5) det (La~) . 0 (;~ row index, 2 column index I \z, 2=t,2 ..... 2p ] Then, /or T su//iciently large, there is also a unique set o/parameters, ~x=fl~ (T), such that L is o] trigonometric order p relative to period T. Furthermore, (1.6) &(r)-+~~ as r~ oo. Pro@ The main difficulty in the proof is the fact that in the limit, as T-+oc, equations (1.3) degenerate into one single equation, L t =0. We therefore trans- form (t.3) into an equivalent set of equations from which the behavior of the solution at T= oo can be studied more easily. In this connection the following trigonometric identities are helpful, r (1.7) sin2'~ = ~r -- cos r x) (r= t,2,3 .... ), 0=1 where ~,0 are suitable real numbers and r The existence of such numbers is obvious, if one observes that sin~' 2 =E(t--cosx)/21' can be written as a cosine-polynomial of exact order r. Differentiating both sides in (1.7) gives also (t.8) sin~r_l x cos-~- = 2 %o sin 0 x (r = t 2, 3 .... ) 2 2 ' ' o=1 where %0= Q~r~o/r, and in particular %,=cU, 4= 0. Equations (t.3) are equivalent to LI=0, L t--cosr-~t2~ ) =Lsmr. 2:rtT-t=O (r=l,2 ..... p). Because of assumption (i) the first of these equations is automatically satisfied. The remaining equations are equivalent to (1.9) a, oL t--cos~T~t = %oLsin~-T-t=0 (r=l,2 ..... p). 0=1 0=1 Numerical integration of ordinary differential equations 383 Using (t.7) and the linearity of L we have 2~ t)= L 2 a~o(l_ cos ~ T ,)_, [ ~, T ] " 0=1 0=1 Similarly, using (t.8), we find 2~ 2zrQLsin r Q=I Therefore, letting (t.10) u= ~ T' we can write (1.9), after suitable multiplications, as follows: L [( sinut)2r-lcosut} =0 (1.tl) (r= 1, 2 ..... p). This represents a system of 2p linear algebraic equations in the same number of unknowns/5~, the coefficient matrix and known vector of which both depend on the parameter u. We show that in the limit as u-~0 the system (t.tl) goes over into the system of equations Lt'=O (r=t, 2 ..... 2p). In fact, it is readily seen, by expansion or otherwise, that for any integers a~O, r>_t, as u--~O, da [( sin'~__ t )2r-l cos u ,] __> d~d a t2r-1, dt a d a (si~t) 2' d ~ ,~r, dt ~ .... § ~d~b- the convergence being uniform with respect to t in any finite interval. In par- ticular, ( sinu t ) 2r-1 /2r-1 cos u t -- --~ 0 (~ ~ 0), II \( sinuu~t/2"--1 ti" -+ 0 so that, by the continuity of the L a, also L sinut 2r--1 t2r_ 1 ~[(~ ) c~ L ~( sinutu )2r ->L~/2' From this our assertion follows immediately. Since the limiting system, by assumption, has a unique solution, flo, the matrix of the system (t.1t) is nonsingular for u=0, and hence remains so for u sufficiently small. It follows that for sufficiently large T there is a unique solution, fla(T), of (1.1t), satisfying (1.6). Theorem I is proved. 384 WALTER GAUTSCHI: Remark 1. Assumption (i) in Theorem t is not essential, but convenient for some of tile applications made later. The theorem holds without the assumption (i) if the functional L in (t.4) is made to depend on 2p+t parameters, (1.4') L x =/30 L o x +/51 L1 x +... + fl2p L2p x + L2p+l x, and assumption (t.5) is modified, accordingly, to (~ row index, ~ column index) (t.5') det (L~t~) ~ 0 \~,2=0, t ..... 2p " The proof remains the same. Remark 2. For particular choices of the La it may happen that the functional L can be made of higher algebraic order than the number of parameters would indicate. Even if the excess in order is a multiple of 2, this does not mean neces- sarily that a similar increase in trigonometric order is possible. For example, Lx=/5 x(0)+ x(t)--kx'(O)--~x'O), /5=--1 if of algebraic order 2, but in general cannot be made of trigonometric order 1, since 2. Linear multi-step methods Linear functionals in C 1 play an important r61e in the numerical solution of first order differential equations (2.t) x' = ! (t, x), x (to) = Xo, in that they provide the natural mathematical setting for a large class of numerical methods, the so-called linear multi-step methods. These are methods which define approximations x,~ to values x (to + mh) of the desired solution by a relation of the following form x~+ 1 + ~1 x~ +.-. + ~ x~+l_ k = h (/50 x'~+l +/51 x', +"" + & x:+l_k) (2.2) (n=k-- l,k,k + l .... ), where x~, =- ! (to + m h, xm). Once k "starting" values xo, x 1 ..... xk_ 1 are known, (2.2) is used to obtain successively all approximations x,~ (m_> k) desired. The integer k> 0 will be called the index of the multi-step method, assuming, of course, that not both ~k and/sk vanish. (2.2) is called an extrapolation method if/5o=-0, and an interpolation method if /5o =4= 0. Interpolation methods require the solution of an equation at each stage, because x'~+l in (2.2) is itself a function of the new approximation x~+ 1. It is natural to associate with (2.2) the linear functional k (2.3) gx=~',I~X(to+(n+l--2)h)--hfl~x'(to+(n+t--~)h)l (~0= t). A=0 The multi-step method (2.2) is called of algebraic order p, if its associated linear functional (2.3) is of algebraic order p; similarly one defines trigonometric order of a multi-step method. Numerical integration of ordinary differential equations 385 Since any linear transformation t' = at+ b (a =~ 0) of the independent variable transforms an algebraic polynomial of degree =<p into one of the same kind, it is clear that (2.2) is of algebraic order p if and only if the functional k (2.4) L 1 x = F, [~ x (k -- 4) -- fl~ x'(k -- 4)] A--0 is of algebraic order p. Here, the parameter h has dropped out, so that the coefficients ~, fl~ of a multi-step method of algebraic order do not depend on h. The situation is somewhat different in the trigonometric case, where a linear transformation other than a translation (or refiexion) changes the period of a trigonometric polynomial.
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