High-frequency wave propagation by the segment projection method
Journal of Computational Physics
Journal of Computational Physics
A semiclassical transport model for two-dimensional thin quantum barriers
Journal of Computational Physics
High order numerical methods to a type of delta function integrals
Journal of Computational Physics
High Order Numerical Quadratures to One Dimensional Delta Function Integrals
SIAM Journal on Scientific Computing
SIAM Journal on Numerical Analysis
A Hybrid Phase-Flow Method for Hamiltonian Systems with Discontinuous Hamiltonians
SIAM Journal on Scientific Computing
Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation
Journal of Computational Physics
| Hi-index | 0.01 |
The phase flow method, originally introduced in [L. X. Ying and E. J. Candés, J. Comput. Phys., 220 (2006), pp. 184-215], can efficiently solve autonomous ordinary differential equations. In [S. Jin, H. Wu, and Z. Y. Huang, SIAM J. Sci. Comput., 31 (2008), pp. 1303-1321], the method was generalized to solve Hamiltonian system where the Hamiltonian function was discontinuous. However, both of these methods require a phase flow map constructed on an invariant manifold. This can increase computational cost when the invariant domain is big or unbounded. Following the idea of [S. Jin, H. Wu, and Z. Y. Huang, SIAM J. Sci. Comput., 31 (2008), pp. 1303-1321], we propose a hybrid phase flow method for solving the Liouville equation in a bounded domain, which is smaller than the invariant manifold of a phase flow map. By using some proper boundary conditions, this method can help solve the problem where the invariant manifold of a phase flow map determined by the Liouville equation is unbounded. We verify numerical accuracy and efficiency by several examples of the semiclassical limit of the Schrödinger equation. Analysis of numerical stability and convergence is given for the semiclassical limit equation with inflow boundary condition.