The Numerical Preservation of Multiple First Integrals of Dynamical Systems

Abstract

Geometric numerical integrators are numerical methods for differential equa tions with invariants such that the numerical solutions preserve the geometric properties or invariants like energy, integrals, symplectic structure, phase-space volume and symmetries. An ordinary differential equation may contain linear, quadratic, cubic or higher order first integrals. For the preservation of quadratic f irst integrals, symplectic numerical method is an appropriate candidate. If we want to preserve cubic or higher order first integrals, we use discrete gradient method and projection method. Discrete gradient method lies in a particular area of geometric numerical integra tion known as integral preserving integrators. Integral preserving integrators are numerical integration algorithms that preserves one or more first integrals of a differential equation. The integrals include energy, momentum and angular mo mentum. The differential equations include Hamiltonian or non-Hamiltonian ones. We consider harmonic oscillator and simple pendulum from Hamiltonian class be cause these are having the differential equations with quadratic first integrals. On the other hand from non-Hamiltonian class, we consider Lorenz and SIRI models because they are having the differential equations with cubic or higher order first integrals. SIRI model is basically a general model of disease transmission with a constant population size N in which we analyze the effect of any disease on a special commu nity. The under observation population is divided into three sections: susceptible, infected, and recovered individuals, labeled as S, I, R. It was firstly formulated by Tudor. The first integrals of SIRI model was calculated by Rehana Naz et al in their research work [8]. They used the partial Hamiltonian approach to obtain the first integrals. In the present work, we firstly studied the discrete gradient method which is a ge ometric numerical integrator for the preservation of first integral of the differential equations. Secondly, we have applied the discrete gradient method on Hamilto ix nian system with the quadratic first integral and report good energy preservation. Moreover, we have applied the discrete gradient method to solve the Lorenz and SIRI model and report good preservation of polynomial first integral of degree greater than two