High Order Numerical Methods to Three Dimensional Delta Function Integrals in Level Set Methods
SIAM Journal on Scientific Computing
A Hybrid Phase Flow Method for Solving the Liouville Equation in a Bounded Domain
SIAM Journal on Numerical Analysis
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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 Candès [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.