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Publications (9)
The Eulerian Gaussian beam method is an efficient way to compute high frequency wave propagation, which was originally studied in Leung et al. (2007) [17]. Later Jin, Wu and Yang developed a new way of computing the Hessian functions from the derivatives of the level set functions in Jin et al. (2008) [19], which greatly reduced the number of equat...
The Dirac equation is an important model in relativistic quantum mechanics. In the
semi-classical regime "≪1, even a spatially spectrally accurate time splitting method [6] requires
the mesh size to be O("), which makes the direct simulation extremely expensive. In this paper, we
present the Gaussian beam method for the Dirac equation. With the hel...
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...
As an important model in quantum semiconductor devices, the Schrodinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for hig...
The linear Schrödinger equation with periodic potentials is an important model in solid
state physics. The most efficient direct simulation using a Bloch decomposition-based
time-splitting spectral method [18] requires the mesh size to be O(epsilon) where epsilon is the scaled
semiclassical parameter. In this paper, we generalize the Gaussian beam...
A novel Eulerian Gaussian beam method was developed in [8] to compute the Schrödinger equation efficiently in the semiclassical regime. In this paper, we introduce an efficient semi-Eulerian implementation of this method. The new algorithm inherits the essence of the Eule-rian Gaussian beam method where the Hessian is computed through the derivativ...
As an important model in quantum semiconductor devices, the Schrödinger-Poisson equations have generated widespread interests in both analysis and numerical simulations in recent years. In this paper, we present Gaussian beam methods for the numerical simulation of the one-dimensional Schrodinger-Poisson equations. The Gaussian beam methods for hig...
The solution to the Schrödinger equation is highly oscillatory when the rescaled
Planck constant $\varepsilon$ is small in the semiclassical regime. A direct numerical simulation requires the
mesh size to be $\emph{O}(\varepsilon)$. The Gaussian beam method is an efficient way to solve the high frequency
wave equations asymptotically, outperformin...
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 (23), the phase map should be smooth to ensure the accuracy of the interpolation. Such an in- terpolation is inaccurate if the phase map is nonsmooth, for example, when the Hamilt...
