A geometric optics method for high-frequency electromagnetic fields computations near fold caustics—Part II. The energy

Benamou, Jean-David; Lafitte, Olivier; Solliec, Ian; Sentis, Rémi

doi:10.1016/j.cam.2003.09.046

Cited by 16 publications

(17 citation statements)

References 6 publications

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“…It is the geometric quantity that measures the evolution of the cross section of an elementary ray tube. This result generalizes, in the case of non-uniform and non-potential mean flow, a well known result for the wave equation [3], and it will be summarized in Proposition 3.4.…”

Section: Computation Of the Geometrical Spreadingsupporting

confidence: 77%

“…The latter is the key quantity for computing the geometrical optics approximation of the acoustic perturbation. Hence, we obtain a generalization to a system of the results of Benamou et al [2,3] for a wave equation with non-constant sound velocity.…”

Section: Resultsmentioning

confidence: 84%

“…Another original feature of this article, from a numerical point of view, is the choice of an Eulerian method for solving both the eikonal equation on the phase φ and the transport equation for the stretching matrix which is a more general tool from what is already used in [3]. The latter is the key quantity for computing the geometrical optics approximation of the acoustic perturbation.…”

Section: Resultsmentioning

confidence: 99%

“…This approach to determine an approximation of the solution of Hamilton-Jacobi equations has been used in several areas of applications, such as in electromagnetic in the high frequency regime (see Benamou et al [2]), the shape-from-shading problem (see Lions, Rouy, Tourin [34]), and optimal control (Crandall, Lions [7]). Note that it differs from the usual ray method, which consists in solving ordinary differential equations for unknowns along a Lagrangian trajectory.…”

Section: Numerical Scheme For the Eikonal Equationmentioning

confidence: 99%

“…This generalizes a known result of the high frequency acoustics in the absence of flow in the high frequency regime (see Theorem 2.11). We do not use this conservative transport equation for the numerical resolution but rather an equivalent approach using the geometrical spreading introduced in a natural fashion by Benamou et al [3] for the wave equation, and which appears to be the key tool also in this more general situation. It is straightforward to deduce from this geometrical spreading the acoustic pressure and the acoustic velocity perturbations.…”

Section: Introductionmentioning

confidence: 99%

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Numerical Eulerian method for linearized gas dynamics in the high frequency regime

2013

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We study the propagation of an acoustic wave in a moving fluid in the high frequency regime. We calculate a high-frequency approximation of the solution of this problem using an Eulerian method.The model retained is a linearized Euler system around a mean fluid flow. For any regular mean flow, we derive a conservative transport equation for the geometrical optics approximation. We introduce the stretching matrix corresponding to this system, from which we deduce the geometrical spreading, key tool for computing the geometrical optics approximation.Finally, we construct and implement a numerical scheme in the Eulerian framework for the eikonal equation. This Eulerian formulation applies also for the transport equation on the stretching matrix.

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Section: Computation Of the Geometrical Spreadingsupporting

confidence: 77%

Section: Resultsmentioning

confidence: 84%

Section: Resultsmentioning

confidence: 99%

Section: Numerical Scheme For the Eikonal Equationmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Numerical Eulerian method for linearized gas dynamics in the high frequency regime

2013

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A Eulerian Geometric Optics Method for High Frequency Electromagnetic Fields Computations in Presence of Fold Caustics

Benamou

Lafitte²,

Sentis

et al. 2003

Mathematical and Numerical Aspects of Wave Propagation WAVES 2003

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A semi-lagrangian numerical method for geometric optics type problems

Benamou

2006

Bit Numer Math

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Dedicated to Björn Engquist on the occasion of his 60th birthday. Abstract.We explore an intermediate "Semi-Lagrangian" formulation of Geometric Optics type problems which stands between the classical "Lagrangian" Hamiltonian systems and its "Eulerian" Hamilton-Jacobi Partial Differential Equations counterpart. The goal is to design a numerical method which "trades" between advantages and drawbacks of both worlds to compute numerically the so-called "multi-valued" solution. We present a numerical algorithm and apply it to the "paraxial" simplification of 2-D geometric optics. We discuss its limitations and its possible extensions to higher dimensions. For an extended discussion on Lagrangian versus Eulerian methods in Geometric Optics and a review of some applications of these techniques see Engquist and Runborg, Acta Numer. (1 The Lagrangian/Eulerian setting. Our prototype problem is the following Hamiltonian system ẏ(s, y o ) = H p (s, y(s, y o ), p(s, y o )), y(0, y o ) = y o , p(s, y o ) = −H y (s, y(s, y o ), p(s, y o )), p(0, y o ) = p o (y o ), (1.1)where y o ∈ R d , s is a "time-like" dimension and( .) = d(.)ds . The notation f α will systematicaly denote partial differentiation of f with respect to α (and therefore a gradient when α ∈ R d with d > 1). The couples (y, p) are called "bicharacteristics" and lie in phase-space R s × R d y × R d p . The curve y(s, y o ) describes the trajectory of a ray issued from point y o and parameterized by s while p(s, y o )

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A geometric optics method for high-frequency electromagnetic fields computations near fold caustics—Part II. The energy

Cited by 16 publications

References 6 publications

Numerical Eulerian method for linearized gas dynamics in the high frequency regime

Numerical Eulerian method for linearized gas dynamics in the high frequency regime

A Eulerian Geometric Optics Method for High Frequency Electromagnetic Fields Computations in Presence of Fold Caustics

A semi-lagrangian numerical method for geometric optics type problems

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