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

Authors:
Francisco R. Villatoro
Affiliations:
Departamento de Lenguajes y Ciencias de la Computación, E.T.S. Ingenieros Industriales, Universidad de Málaga, Plaza El Ejido, s/n, 29013-Málaga, Spain
Venue:
Journal of Computational and Applied Mathematics
Year:
2009

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Abstract

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 third-order rational methods are presented.