Exact finite-difference methods based on nonlinear means

Abstract

The necessary and sufficient condition for an autonomous, ordinary differential equation to be exactly solved by a given Evans-Sanugi, nonlinear, one-step, finite difference method based on ‘classical means’ are derived. A necessary, but not sufficient, condition yields the most general nonlinearity for the differential equation which is independent of the step size. Examples of differential equations for which either nonlinear trapezoidal or nonlinear implicit midpoint methods based on arithmetic, harmonic, contraharmonic, quadratic, geometric, Heronian, centroidal and logarithmic means are exact, are presented. These new exact difference schemes may be useful in future developments of new Denk-Bulirsch, Le-Roux or Kojouhavov-Chen schemes for nonlinear evolution equations with or without blow-up.

Keywords:

• Evans-Sanugi, nonlinear trapezoidal methods
• Evans-Sanugi, nonlinear, implicit midpoint methods
• exact difference schemes
• nonlinear one-step methods
• classical means

Acknowledgements

The research reported in this paper was supported by Project FIS2005-03191 from the Ministerio de Educación y Ciencia, Spain.