Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation

Jin, Shi; Wen, Xin

doi:10.1016/j.wavemoti.2006.06.001

Cited by 13 publications

(8 citation statements)

References 43 publications

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“…In general, one probably needs to modify slope limiter near the interface by using the Hamiltonian-preserving principle across the interface, as in [21]. For the numerical examples in the next section, we just use the monotonized central-difference limiter, which works well for our examples; therefore we do not consider this issue here.…”

Section: The Numerical Fluxesmentioning

confidence: 99%

An Eulerian Surface Hopping Method for the Schrödinger Equation with Conical Crossings

Jin

Zhang

2011

Multiscale Model. Simul.

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Abstract. In a nucleonic propagation through conical crossings of electronic energy levels, the codimension two conical crossings are the simplest energy level crossings, which affect the BornOppenheimer approximation in the zeroth order term. The purpose of this paper is to develop the surface hopping method for the Schrödinger equation with conical crossings in the Eulerian formulation. The approach is based on the semiclassical approximation governed by the Liouville equations, which are valid away from the conical crossing manifold. At the crossing manifold, electrons hop to another energy level with the probability determined by the Landau-Zener formula. This hopping mechanics is formulated as an interface condition, which is then built into the numerical flux for solving the underlying Liouville equation for each energy level. While a Lagrangian particle method requires the increase in time of the particle numbers, or a large number of statistical samples in a Monte Carlo setting, the advantage of an Eulerian method is that it relies on fixed number of partial differential equations with a uniform in time computational accuracy. We prove the positivity and l 1 -stability and illustrate by several numerical examples the validity and accuracy of the proposed method.

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Section: The Numerical Fluxesmentioning

confidence: 99%

An Eulerian Surface Hopping Method for the Schrödinger Equation with Conical Crossings

Jin

Zhang

2011

Multiscale Model. Simul.

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“…While the Liouville equation (4.1) can be solved by a standard finite difference or finite volume shock capturing method, such schemes face difficulties when the Hamiltonian is discontinuous, since ignoring the discontinuity of the Hamiltonian during the computation will result in solutions inconsistent with the notion of the (physically relevant) solution defined in the preceding subsection. Even with a smoothed Hamiltonian, it is usually impossible-at least in the case of partial transmission and reflection-to obtain the correct amount of transmissions and reflections (see some numerical examples in [19,21]). A smoothed Hamiltonian will also give a severe time step constraint like ∆t ∼ O(∆x∆p), where ∆t, ∆x and ∆p are time step, mesh sizes in the x-and p-directions respectively.…”

Section: 1)mentioning

confidence: 99%

“…This is the first Eulerian numerical methods for high frequency waves that are able to capture correctly the transmission and reflection of waves through the barriers or interfaces. The framework also extends to reduced Liouville equation [21], where the interface condition uses directly Snell's Law to determine one side of the wave direction vector from the other side. It has also been extended to high frequency elastic waves [12], and high frequency waves in random media [13] with diffusive interfaces, where radiative transfer equation rather than the Liouville equation was used.…”

Section: 1)mentioning

confidence: 99%

Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond

Jin¹

2009

Hyperbolic Problems: Theory, Numerics and Applications

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Abstract. This paper reviews some recent numerical methods for hyperbolic equations with singular (discontinuous or measure-valued) coefficients. Such problems arise in wave propagation through interfaces or barriers, or nonlinear waves through singular geometries. The connection between the well-balanced schemes for shallow-water equations with discontinuous bottom topography and the Hamiltonian preserving schemes for Liouville equations with discontinuous Hamiltonians is illustrated. Various developments of numerical methods with applications in high frequency waves through interfaces, and in multiscale coupling between classical and quantum mechanics, are discussed..

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“…It is mainly because the slope along the interface is just zero in those examples. In general, one probably needs to modify slope limiter near the interface, again by using the Hamiltonian preserving principle across the interface, as in [13].…”

Section: Algorithmmentioning

confidence: 99%

A Hamiltonian-Preserving Scheme for High Frequency Elastic Waves in Heterogeneous Media

Jin

Liao

2006

J. Hyper. Differential Equations

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We develop a class of Hamiltonian-preserving 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.

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Computation of transmissions and reflections in geometrical optics via the reduced Liouville equation

Cited by 13 publications

References 43 publications

An Eulerian Surface Hopping Method for the Schrödinger Equation with Conical Crossings

An Eulerian Surface Hopping Method for the Schrödinger Equation with Conical Crossings

Numerical methods for hyperbolic systems with singular coefficients: well-balanced scheme, Hamiltonian preservation, and beyond

A Hamiltonian-Preserving Scheme for High Frequency Elastic Waves in Heterogeneous Media

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