Structure Preserving Pseudo-Runge-Kutta Method

Abstract

The present paper is intended to propose pseudo-RungeKutta method (PRKM) which is quadratically invariant i. e. it preserves structural properties when the PRKM is applied to the Hamiltonian system of equations. We have inserted the area preserving character in the implicit pseudo-Runge-Kutta method and derived the sufficient conditions of symplecticness for the pseudo-Runge-Kutta method and thus developed a qualitative numerical method. These methods are best tuned to solve system of partial differential equations of Hamiltonian type. Though these methods are not self starting but the order of the truncation error is equivalent to its counterpart Runge-Kutta method. These methods can be used to solve numerically the dynamical system of equations of Hamiltonian type such that the Hamiltonian is preserved in the numerical solution. The derivation of sufficient conditions is based on differential forms.