In this paper a fast sweeping method for computing the numerical solution of Eikonal equations on a rectangular grid is presented. The method is an iterative method which uses upwind difference for discretization and uses Gauss-Seidel iterations with alternating sweeping ordering to solve the discretized system. The crucial idea is that each sweeping ordering follows a family of characteristics...

The paper is devoted to the modelling of the propagation of acoustic waves at high frequency in anisotropic media. In this case, the WKB approximation results in eikonal equations whose Hamiltonians are neither convex nor concave in the impulse variable so that the Fermat principle fails. In this paper, methods of differential games are adopted for the analysis of wave propagation. If the Hamil...

We present the ultraviolet poles and finite terms of two-loop vertex corrections for diagrams with massive partons in the eikonal approximation. We discover and prove that the results for a set of the corrections generalize to n-loop eikonal diagrams. These results will enhance theoretical understanding and allow greater calculational accuracy for the many partonic processes where the eikonal a...

Single-particle dynamics in electron microscopes, ion or electron lithographic instruments, particle accelerators, and particle spectrographs is described by weakly nonlinear ordinary differential equations. Therefore, the linear part of the equation of motion is usually solved and the nonlinear effects are then found in successive order by iteration methods. When synchrotron radiation is not i...

S U M M A R Y In this paper, several methods for the solution of the eikonal equation are investigated. Using the method of characteristics, traveltimes are computed along rays. Approximate solutions of the eikonal equation can also be obtained. From Fermat’s principle, first-order changes in the traveltime can be found by integrating along initial ray trajectories. Alternatively, linearized ra...

We present a method of ray tracing that is based on a system of differential equations equivalent to the eikonal equation, but formulated in the ray coordinate system. We use a first-order discretization scheme that is interpreted very simply in terms of the Huygens’ principle. The method has proved to be a robust alternative to conventional ray tracing, while being faster and having a better a...

We first derive the relation between the eikonal approximation to the Maxwell wave equations in an inhomogeneous anisotropic medium and geodesic motion in a three dimensional Riemannian manifold using a method which identifies the symplectic structure of the corresponding mechanics. We then apply an analogous method to the five dimensional generalization of Maxwell theory required by the gauge ...

The use of differential equations such as Eikonal, Hamilton-Jacobi and Poisson for the economical calculation of the nearest wall distance d, which is needed by some turbulence models, is explored. Modifications that could palliate some turbulence-modeling anomalies are also discussed. Economy is of especial value for deforming/adaptive grid problems. For these, ideally, d is repeatedly compute...

A bstract Understanding the spin structure of hadrons in small x regime is an important direction to unravel puzzle hadronic physics. To include degrees freedom requires going beyond usual eikonal approximation high energy QCD. We developed effective Hamiltonian approach study related observables using shockwave formalism. The small- incorporates both quark and gluon propagators background fiel...