Structure-preserving algorithms for dynamical systems

We study structure-preserving algorithms to phase space volume for linear dynamical systems y(over dot) = Ly for which arbitrarily high order explicit symmetric structure-preserving schemes,i.e. the numerical solutions generated by the schemes satisfy det(partial derivative(y1)/partial derivative(y0)) = e(htrL) where trL is the trace of matrix L, can be constructed. For nonlinear dynamical systems y(over dot) = 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.