Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation

Jin, Shi; Liu, Hailiang; Osher, Stanley; Tsai, Yen‐Hsi Richard

doi:10.1016/j.jcp.2004.11.008

Cited by 70 publications

(135 citation statements)

References 33 publications

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“…Beyond caustics, the correct semiclassical limit of the Schrödinger equation becomes multivalued. The multivalued solution can be computed by ray tracing methods [6,2,3], wave front methods [32,9], moment methods [11,14] and level set methods [16,5,15,17]. We also refer the readers to the review paper on computational high frequency waves [8].…”

Section: )mentioning

confidence: 99%

“…This new formulation significantly reduces the number of Liouville equations used to construct the Hessian of the phase. As a matter of fact, the computational method for the (complex-valued) phase and amplitude is not much different from the level set method used for geometric optics computations as in [5,16,15]. In addition, we also evaluate the Gaussian beam summation integral using the semi-Lagrangian method of [19] only near caustics, thus largely maintain the accuracy of the Eulerian method.…”

Section: )mentioning

confidence: 99%

“…In the last few years, the level set method has been developed to compute the multi-valued solution of (1.7)-(1.8) which gives the correct semiclassical limit of the Schrödinger solution [16,5,15] away from the caustics. The idea is to build the velocity u = ∇ y S into the intersection of zero level sets of phase-space functions φ j (t,y,ξ), j = 1,··· ,n, i.e.,…”

Section: The Eulerian Formulationmentioning

confidence: 99%

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Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

Jin

Yang

2008

Communications in Mathematical Sciences

129

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Abstract. 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 (complex-valued) 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.

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: The Eulerian Formulationmentioning

confidence: 99%

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Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

Jin

Yang

2008

Communications in Mathematical Sciences

129

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“…We make no attempt to comment on these methods, but refer to [21] for a seminar survey on computational high-frequency wave propagation. More recently, a geometric view point has been adopted in place of the kinetic one in phase space; here we shall outline the corresponding level set methods developed in [8,[35][36][37]51,52]. Traditionally the level set method has been a highly successful computational technique for capturing the evolution of curves and surfaces [67,68] with applications in diverse areas such as multi-phase fluids, computer vision, imaging processing, optimal shape design, etc.…”

Section: Level Set Equationmentioning

confidence: 99%

“…Based on the level set framework in the phase space, the amplitude is evaluated by ρ(t, x) = f (t, x, p)δ(φ)dp, (4.13) where the quantity f also solves the same Liouville equation (4.12) but with f (0, x, p) = ρ 0 (x) as initial data. The multi-valued higher moments can be also resolved by integrating f along the bi-characteristic manifold in the phase directions (see [35,36]). We refer the reader to the review article [52] for further details.…”

Section: Level Set Equationmentioning

confidence: 99%

On discreteness of the Hopf equation

Liu

2008

Acta Math. Appl. Sin. Engl. Ser.

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The principle aim of this essay is to illustrate how different phenomena is captured by different discretizations of the Hopf equation and general hyperbolic conservation laws. This includes dispersive schemes, shock capturing schemes as well as schemes for computing multi-valued solutions of the underlying equation. We introduce some model equations which describe the behavior of the discrete equation more accurate than the original equation. These model equations can either be conveniently discretized for producing novel numerical schemes or further analyzed to enrich the theory of nonlinear partial differential equations.

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Regularized Dirac delta functions for phase field models

Lee

Kim

2012

Numerical Meth Engineering

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SUMMARY The phase field model is a highly successful computational technique for capturing the evolution and topological change of complex interfaces. The main computational advantage of phase field models is that an explicit tracking of the interface is unnecessary. The regularized Dirac delta function is an important ingredient in many interfacial problems that phase field models have been applied. The delta function can be used to postprocess the phase field solution and represent the surface tension force. In this paper, we present and compare various types of delta functions for phase field models. In particular, we analytically show which type of delta function works relatively well regardless of whether an interfacial phase transition is compressed or stretched. Numerical experiments are presented to show the performance of each delta function. Numerical results indicate that (1) all of the considered delta functions have good performances when the phase field is locally equilibrated; and (2) a delta function, which is the absolute value of the gradient of the phase field, is the best in most of the numerical experiments. Copyright © 2012 John Wiley & Sons, Ltd.

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Computing multivalued physical observables for the semiclassical limit of the Schrödinger equation

Cited by 70 publications

References 33 publications

Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

Gaussian beam methods for the Schrodinger equation in the semi-classical regime: Lagrangian and Eulerian formulations

On discreteness of the Hopf equation

Regularized Dirac delta functions for phase field models

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