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GAUSSIAN BEAM METHODS FOR THE SCHRÖDINGER EQUATION IN THE SEMICLASSICAL REGIME: LAGRANGIAN AND EULERIAN FORMULATIONS
, 2008
"... The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant ε is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be O(ε). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptoti ..."
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Cited by 29 (12 self)
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The solution to the Schrödinger equation is highly oscillatory when the rescaled Planck constant ε is small in the semiclassical regime. A direct numerical simulation requires the mesh size to be O(ε). The Gaussian beam method is an efficient way to solve the high frequency wave equations asymptotically, outperforming the geometric optics method in that the Gaussian beam method is accurate even at caustics. In this paper, we solve the Schrödinger equation using both the Lagrangian and Eulerian formulations of the Gaussian beam methods. A new Eulerian Gaussian beam method is developed using the level set method based only on solving the (complexvalued) homogeneous Liouville equations. A major contribution here is that we are able to construct the Hessian matrices of the beams by using the level set function’s first derivatives. This greatly reduces the computational cost in computing the Hessian of the phase function in the Eulerian framework, yielding an Eulerian Gaussian beam method with computational complexity comparable to that of the geometric optics but with a much better accuracy around caustics. We verify through several numerical experiments that our Gaussian beam solutions are good approximations to Schrödinger solutions even at caustics. We also numerically study the optimal relation between the number of beams and the rescaled Planck constant ε in the Gaussian beam summation.
A Semiclassical transport model for thin quantum barriers, Multiscale Modeling and Simulation
 Simul
, 2006
"... Abstract. We present a onedimensional timedependent semiclassical transport model for mixed state scattering with thin quantum barriers. The idea is to solve a stationary Schrödinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interfa ..."
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Cited by 19 (15 self)
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Abstract. We present a onedimensional timedependent semiclassical transport model for mixed state scattering with thin quantum barriers. The idea is to solve a stationary Schrödinger equation in the thin quantum barrier to obtain the scattering coefficients, and then use them to supply the interface condition that connects the two classical domains. We then build in the interface condition to the numerical flux, in the spirit of the Hamiltonianpreserving scheme introduced by Jin and Wen for a classical barrier. The overall cost is roughly the same as solving a classical barrier. We construct a numerical method based on this semiclassical approach and validate the model using various numerical examples.
A Hamiltonianpreserving scheme for high frequency elastic waves inheterogeneous media
 J. Hyperbolic Diff Eqn
"... We develop a class of Hamiltonianpreserving numerical schemes for high frequency elastic waves in heterogeneous media. The approach is based on the high frequency approximation governed by the Liouville equations with singular coefficients due to material interfaces. As previously done by Jin and W ..."
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Cited by 13 (11 self)
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We develop a class of Hamiltonianpreserving numerical schemes for high frequency elastic waves in heterogeneous media. The approach is based on the high frequency approximation governed by the Liouville equations with singular coefficients due to material interfaces. As previously done by Jin and Wen [10, 12], we build into the numerical flux the wave scattering information at the interface, and use the Hamiltonian preserving principle to couple the wave numbers at both sides of the interface. This gives a class of numerical schemes that allows a hyperbolic CFL condition, is positive and l ∞ stable, and captures correctly wave scattering at the interface with a sharp numerical resolution. We also extend the method to curved interfaces. Numerical experiments are carried out to study the numerical convergence and accuracy. 1
Computational high frequency waves through curved interfaces via the Loiuville equation and Geometric Theory of Diffraction
 J. Comput. Phys
, 2008
"... We construct a class of numerical schemes for the Liouville equation of geometric optics coupled with the Geometric Theory of Diffractions to simulate the high frequency linear waves with a discontinuous index of refraction. In the work [26], a Hamiltonianpreserving scheme for the Liouville equatio ..."
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Cited by 11 (8 self)
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We construct a class of numerical schemes for the Liouville equation of geometric optics coupled with the Geometric Theory of Diffractions to simulate the high frequency linear waves with a discontinuous index of refraction. In the work [26], a Hamiltonianpreserving scheme for the Liouville equation was constructed to capture partial transmissions and reflections at the interfaces. These schemes are extended by incorporating diffraction terms derived from Geometric Theory of Diffraction into the numerical flux in order to capture diffraction at the interface. We give such a scheme for curved interfaces. This scheme is proved to be positive under a suitable time step constraint. Numerical experiments show that it can capture diffraction phenomena without fully resolving the wave length of the original wave equation. 1
Convergence of an immersed interface upwind scheme for linear advection equations with piecewise constant coefficients II: Some related binomial coefficients inequalities
"... We study the L1error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the nu ..."
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Cited by 10 (9 self)
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We study the L1error estimates for the upwind scheme to the linear advection equations with a piecewise constant coefficients modeling linear waves crossing interfaces. Here the interface condition is immersed into the upwind scheme. We prove that, for initial data with a bounded variation, the numerical solution of the immersed interface upwind scheme converges in L1norm to the differential equation with the corresponding interface condition. We derive the onehalfth order L1error bounds with explicit coefficients following a technique used in [25]. We also use some inequalities on binomial coefficients proved in a consecutive paper [32].
A coherent semiclassical transport model for purestate quantum scattering
 Comm. Math. Sci
, 2009
"... We present a timedependent semiclassical transport model for coherent purestate scattering with quantum barriers. The model is based on a complexvalued Liouville equation, with interface conditions at quantum barriers computed from the steadystate Schrödinger equation. By retaining the phase inf ..."
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Cited by 8 (6 self)
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We present a timedependent semiclassical transport model for coherent purestate scattering with quantum barriers. The model is based on a complexvalued Liouville equation, with interface conditions at quantum barriers computed from the steadystate Schrödinger equation. By retaining the phase information at the barrier, this coherent model adequately describes quantum scattering and interference at quantum barriers, with a computational cost comparable to that of the classical mechanics. We construct both Eulerian and Lagrangian numerical methods for this model, and validate it using several numerical examples, including multiple quantum barriers. 1
Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation
 WAVE MOTION
, 2006
"... We develop a numerical scheme for the wave front computation of complete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear w ..."
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Cited by 8 (8 self)
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We develop a numerical scheme for the wave front computation of complete transmissions and reflections in geometrical optics. Such a problem can be formulated by a reduced Liouville equation with a discontinuous local wave speed or index of refraction, arising in the high frequency limit of linear waves through inhomogeneous media. The key idea is to incorporate Snell’s Law of Refraction into the numerical flux for the reduced Liouville equation. This scheme allows a hyperbolic CFL condition, under which positivity, and stabilities in both l∞ and l¹ norms, are established. Numerical experiments are carried out to demonstrate the validity and accuracy of this new scheme.
The L 1 stability of a Hamiltonianpreserving scheme for the Liouville equation with discontinuous potentials, preprint
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A domain decomposition method for a twoscale transport equation with energy flux conserved at the interface, Kinetic and Related Models
 with X. Yang and G.W. Yuan) , Kinetic and Related Models
, 2008
"... Abstract. When a linear transport equation contains two scales, one diffusive and the other nondiffusive, it is natural to use a domain decomposition method which couples the transport equation with a diffusion equation with an interface condition. One such method was introduced by Golse, Jin and L ..."
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Cited by 6 (5 self)
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Abstract. When a linear transport equation contains two scales, one diffusive and the other nondiffusive, it is natural to use a domain decomposition method which couples the transport equation with a diffusion equation with an interface condition. One such method was introduced by Golse, Jin and Levermore in [11], where an interface condition, which is derived from the conservation of energy density, was used to construct an efficient noniterative domain decomposition method. In this paper, we extend this domain decomposition method to diffusive interfaces where the energy flux is conserved. Such problems arise in high frequency waves in random media. New operators corresponding to transmission and reflections at the interfaces are derived and then used in the interface conditions. With these new operators we are able to construct both first and second order (in terms of the mean free path) noniterative domain decomposition methods of the type by GolseJinLevermore, which will be proved having the desired accuracy and tested numerically. 1. Introduction. The
Computation of High Frequency Wave Diffraction by a Half Plane via the Liouville Equation and Geometric Theory of Diffraction
 COMMUN COMPUT PHYS
, 2008
"... We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a half plane. We first introduce a condition, based on the GTD theory, at the vertex of the half plan ..."
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Cited by 5 (4 self)
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We construct a numerical scheme based on the Liouville equation of geometric optics coupled with the Geometric Theory of Diffraction (GTD) to simulate the high frequency linear waves diffracted by a half plane. We first introduce a condition, based on the GTD theory, at the vertex of the half plane to account for the diffractions, and then build in this condition as well as the reflection boundary condition into the numerical flux of the geometrical optics Liouville equation. Numerical experiments are used to verify the validity and accuracy of this new Eulerian numerical method which is able to capture the moments of high frequency and diffracted waves without fully resolving the high frequency numerically.