Structure-Preserving Algorithms for Dynamical Systems

Abstract

We study structure-preserving algorithms to phase space volume for linear dynamical systems $\dot{y} = Ly$ for which arbitrarily high order explicit symmetric structure-preserving schemes, i.e. the numerical solutions generated by the schemes satisfy $\det(\frac{\partial y_1}{\partial y_0})=e^{htrL}$, where $trL$ is the trace of matrix $L$, can be constructed. For nonlinear dynamical systems $\dot{y}=f(y)$ Feng-Shang first-order volume-preserving scheme can be also constructed starting from modified $\theta-$methods and is shown that the scheme is structure-preserving to phase space volume.