Construction of an extended invariant for an arbitrary ordinary differential equation with its development in a numerical integration algorithm

Abstract

For an arbitrary ordinary differential equation (ODE), a scheme for constructing an extended ODE endowed with a time-invariant function is here proposed. This scheme enables us to examine the accuracy of the numerical integration of an ODE that may itself have had no invariant. These quantities are constructed by referring to the Nosé-Hoover molecular dynamics equation and its related conserved quantity. By applying this procedure to several molecular dynamics equations, the conventional conserved quantity individually defined in each dynamics can be reproduced in a uniform, generalized way; our concept allows a transparent outlook underlying these quantities and ideas. Developing the technique, for a certain class of ODEs we construct a numerical integrator that is not only explicit and symmetric, but preserves a unit Jacobian for a suitably defined extended ODE, which also provides an invariant. Our concept is thus to simply build a divergence-free extended ODE whose solution is just a lift-up of the original ODE, and to constitute an efficient integrator that preserves the phase-space volume on the extended system. We present precise discussions about the general mathematical properties of the integrator and provide specific conditions that should be incorporated for practical applications.