Given a one-step numerical scheme, on which ordinary differential equations is it exact?

Abstract

MSC: 39A05 65L05Keywords: Nonstandard finite difference methods Rational one-step methods Exact difference methods a b s t r a c t A necessary condition for a (non-autonomous) ordinary differential equation to be exactly solved by a one-step, finite difference method is that the principal term of its local truncation error be null. A procedure to determine some ordinary differential equations exactly solved by a given numerical scheme is developed. Examples of differential equations exactly solved by the explicit Euler, implicit Euler, trapezoidal rule, second-order Taylor, third-order Taylor, van Niekerk's second-order rational, and van Niekerk's thirdorder rational methods are presented.

Contributions to the development of differential systems exactly solved by multistep finite-difference schemes

“…These values of y n+1 coincide with each of the corresponding exact solutions evaluated at x n + h (for details see [6] and [8]), except for the differential equation in (9). The resulting differential equations for the Trapezoidal Rule are valid for any other method for which the local truncation error is similar to that in (6), and also for the second-order Taylor method, and for the two-step Adams-Bashforth method in Section 2.3.…”

“…We observe that, in view of the fact that the C i are arbitrary constants, the first three differential equations in (7) and the differential equation in (9), are the same as those in Section 3.2 of [6].…”

“…as presented in [6], or as previously appeared in [14]. In fact, the procedure presented in [13] for obtaining one-step methods for solving I.V.P.s leads in the simpler case to the first method of the family of rational one-step methods of van Niekerk [14].…”

“…Evidently, a necessary condition for an I.V.P. of the form in (1) to be solved exactly by the method in (2) is that its solution must solve the ordinary differential equation given by LðyÞ ¼ 0 (see [6] for the case of one-step methods). That is, the differential equations formed by the solutions of LðyÞ ¼ 0 are candidates to be solved exactly by the numerical method.…”

“…This procedure is the extension of the technique developed in [6] from one-step methods to multistep methods for scalar differential equations. In what follows we will illustrate the procedure considering different examples of discrete numerical methods.…”