Volume-preserving algorithms for source-free dynamical systems (Q1907119)

A source-free dynamical system is defined by the differential equation \(\dot x= a(x)\) with \(\text{div}(a)= 0\). To solve this system numerically for \(x\), it is quite recommended to use volume-preserving algorithms; the aim of the paper is to construct difference schemes preserving this structure. One first shows that the conventional methods, including Runge-Kutta methods and Euler methods are non-volume preserving, and then one defines the ``essentially Hamiltonian decomposition'' of source-free vector fields. This decomposition is then used to construct a volume-preserving difference scheme.