A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain

The phase flow method was originally introduced in [28] which can efficiently compute the autonomous ordinary differential equations. In [13], it was generalized to solve the Hamiltonian system where the Hamiltonian contains discontinuous functions, for example discontinuous potential or wave speed. However, both these works require the flow map constructed on an invariant manifold. This could lead to an expensive computational cost when the invariant domain is big or even unbounded. In this paper, following the idea of [13], we propose a hybrid phaseflow method for solving the Liouville equation in the bounded domain where the flow map sits in the variant manifold of the traditional phase flow map. Moreover, with the help of some proper boundary conditions, this hybrid phase flow method could help reduce the numerical difficulty when the invariant manifold of the phase flow given by the Liouville equation is big or unbounded. Analysis of the numerical stability and convergence is given for the Liouville equation with the inflow boundary condition. We also verify the accuracy and efficiency of this algorithm by several examples related to the semiclassical limit of the Schrödinger equation.