Numerical integration of ordinary differential equations based on trigonometric polynomials

Abstract

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 polynomials 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. Linear functionals of algebraic and trigonometric order .. p)'it is said to be of trigonometric order p, relative to period T, if 2z~ t =Lsln r ~-.t =0 (r=t,2 ..... p).Thus, a functional L is of algebraic order p if it annihilates all algebraic polynomials 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.

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