A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians

In this paper, we propose a new phase-flow method for Hamiltonian systems with discontinuous Hamiltonians. In the original phase-flow method introduced by Ying and Candes [J. Comput. Phys., 220 (2006), pp. 184-215], the phase map should be smooth to ensure the accuracy of the interpolation. Such an interpolation is inaccurate if the phase map is nonsmooth, for example, when the Hamiltonian is discontinuous. We modify the phase-flow method using a discontinuous Hamiltonian solver and establish the stability (for piecewise constant potentials) of such a solver. This extends the applicability of the highly efficient phase-flow method to singular Hamiltonian systems, with a mild increase of algorithm complexity. Such a particle method can be useful for the computation of high frequency waves through interfaces.

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