A new method for solving the eikonal equation in the spherical computing domain

In order to overcome the problem of singularities and nonuniform grids arising when solving eikonal equation in spherical coordinate systems, a spherical Cartesian coordinate system is defined and the Hamiltonian form of the eikonal equation according to this coordinate system is given. A modified velocity function that can transform spherical coordinate system–based eikonal equation into ones based on a spherical Cartesian coordinate system is deduced by using a differential geometric method where a layered distribution of the velocity function is assumed. After comparing the results of using this approach with the traditional method of solving eikonal equation based on a spherical coordinate system, the viability of the transformation to a spherical Cartesian coordinate system based on a modified velocity function is proven. Despite the assumption of a layered distribution of the velocity function, it is also proven that the method will hold for a velocity function under any three-dimensional distribution. The new method overcomes problems present in traditional approaches and opens up a new way of solving eikonal equation in a spherical computational domain.     

Plane wave refraction on convex and concave angles in geometric acoustics approximation

An exact analytical solution to the eikonal equation for a plane wave refracted on a boundary comprising both convex and concave obtuse angles has been built. Under the convex angle summit the solution has a line of discontinuity in the ray vector field and in the first derivatives of the first arrival times, and under the concave angle it has a cone of waves diffracted on this angle. This cone corresponds to the Keller diffraction cone in the geometric diffraction theory. The relation between the eikonal equation and the resultant Hamilton–Jacoby equation for arrival times of downward waves and the ray parameter conservation equation is investigated. Solutions to these equations coincide for pre-critical incidence angles and differ for super-critical angles. It is shown that the arrival times of maximum amplitude waves, which are of the greatest practical interest, coincide with the times calculated from the ray parameter field for the ray parameter conservation equation. The numerical algorithm proposed for calculation of these times can be used for arbitrary speed models.     

Numerical analysis of an inverse problem for the eikonal equation

We are concerned with the inverse problem for an eikonal equation of determining the speed function using observations of the arrival time on a fixed surface. This is formulated as an optimisation problem for a quadratic functional with the state equation being the eikonal equation coupled to the so-called Soner boundary condition. The state equation is discretised by a suitable finite difference scheme for which we obtain existence, uniqueness and an error bound. We set up an approximate optimisation problem and show that a subsequence of the discrete mimina converges to a solution of the continuous optimisation problem as the mesh size goes to zero. The derivative of the discrete functional is calculated with the help of an adjoint equation which can be solved efficiently by using fast marching techniques. Finally we describe some numerical results.     

A numerical algorithm for computing tsunami wave amplitudes

A numerical multistep algorithm for computing tsunami wave front amplitudes is proposed. The first step consists in solving an appropriate eikonal equation. The eikonal equation is solved with Godunov’s approach and a bicharacteristic method. A qualitative comparison of the two methods is done. A change of variables is made with the eikonal equation solution at the second step. At the last step, using an expansion of the fundamental solution to the shallow water equations in the new variables, we obtain a Cauchy problem of lesser dimension for the leading edge wave amplitude. The results of numerical experiments are presented.     

A Study of the Direct Spectral Transform for the Defocusing Davey-Stewartson II Equation the Semiclassical Limit

The defocusing Davey-Stewartson II equation has been shown in numerical experiments to exhibit behavior in the semiclassical limit that qualitatively resembles that of its one-dimensional reduction, the defocusing nonlinear Schrödinger equation, namely the generation from smooth initial data of regular rapid oscillations occupying domains of space-time that become well-defined in the limit. As a first step to studying this problem analytically using the inverse scattering transform, we consider the direct spectral transform for the defocusing Davey-Stewartson II equation for smooth initial data in the semiclassical limit. The direct spectral transform involves a singularly perturbed elliptic Dirac system in two dimensions. We introduce a WKB-type method for this problem, proving that it makes sense formally for sufficiently large values of the spectral parameter k by controlling the solution of an associated nonlinear eikonal problem, and we give numerical evidence that the method is accurate for such k in the semiclassical limit. Producing this evidence requires both the numerical solution of the singularly perturbed Dirac system and the numerical solution of the eikonal problem. The former is carried out using a method previously developed by two of the authors, and we give in this paper a new method for the numerical solution of the eikonal problem valid for sufficiently large k. For a particular potential we are able to solve the eikonal problem in closed form for all k, a calculation that yields some insight into the failure of the WKB method for smaller values of k. Informed by numerical calculations of the direct spectral transform, we then begin a study of the singularly perturbed Dirac system for values of k so small that there is no global solution of the eikonal problem. We provide a rigorous semiclassical analysis of the solution for real radial potentials at k=0, which yields an asymptotic formula for the reflection coefficient at k=0 and suggests an annular structure for the solution that may be exploited when k ≠ 0 is small. The numerics also suggest that for some potentials the reflection coefficient converges pointwise as ɛ↓ 0 to a limiting function that is supported in the domain of k-values on which the eikonal problem does not have a global solution. It is expected that singularities of the eikonal function play a role similar to that of turning points in the one-dimensional theory. © 2019 Wiley Periodicals, Inc.     

Exponentially Localized Solutions to the Klein–Gordon Equation

Exponentially localized solutions of the Klein–Gordon equation for two and three space variables are presented. The solutions depend on four free parameters. For some relations between the parameters, the solutions describe wave packets filled with oscillations whose amplitudes decrease in the Gaussian way with distance from a point running with group velocity along a ray. The solutions are constructed by using exact complex solutions of the eikonal equation and may be regarded as ray solutions with amplitudes involving one term. It is also shown that the multidimensional nonlinear Klein–Gordon equation can be reduced to an ordinary differential equation with respect to the complex eikonal. Bibliography: 12 titles.     

Dynamic Programming, the Fermat Principle, and the Eikonal Equation Revisited

We derive the eikonal equation using the technique of dynamic programming. The formalism of dynamic programming is based on the procedure defining the minimum pathway, and an immediate application of the Fermat principle leads to the well-known eikonal equation of classical geometrical optics.     

The eikonal equation of an indefinite metric

The concept of a semi-Riemannian map is introduced and it is shown that such maps are solutions of the eikonal equation. Also the existence of solutions to the eikonal equation are discussed and their relation to the Laplace-Beltrami equation is investigated.Supported by the project TBAG-CG2, Tübitak, Turkey.     

Convergence results for an inhomogeneous system arising in various high frequency approximations

Summary. This paper is devoted to both theoretical and numerical study of a system involving an eikonal equation of Hamilton-Jacobi type and a linear conservation law as it comes out of the geometrical optics expansion of the wave equation or the semiclassical limit for the Schr?dinger equation. We first state an existence and uniqueness result in the framework of viscosity and duality solutions. Then we study the behavior of some classical numerical schemes on this problem and we give sufficient conditions to ensure convergence. As an illustration, some practical computations are provided. Received December 6, 1999 / Revised version received August 2, 2000 / Published online June 7, 2001     

Macroscopic modeling and simulations of room evacuation

We analyze numerically two macroscopic models of crowd dynamics: the classical Hughes model and the second order model being an extension to pedestrian motion of the Payne–Whitham vehicular traffic model. The desired direction of motion is determined by solving an eikonal equation with density dependent running cost, which results in minimization of the travel time and avoidance of congested areas. We apply a mixed finite volume-finite element method to solve the problems and present error analysis for the eikonal solver, gradient computation and the second order model yielding a first order convergence. We show that Hughes’ model is incapable of reproducing complex crowd dynamics such as stop-and-go waves and clogging at bottlenecks. Finally, using the second order model, we study numerically the evacuation of pedestrians from a room through a narrow exit.     

Propagation of Surface Waves at High Frequencies

An asymptotic theory is presented for the analysis of surfacewave propagation at high frequencies. The theory is developedfor scalar surface waves satisfying an impedance boundary conditionon a surface, which may be curved and, whose impedance may bevariable. A surface eikonal equation is derived for the phaseof the surface wave field, and it is shown that the wave fieldpropagates over the surface along the surface rays, which arethe characteristics of the surface eikonal equation. The wavefield in space is found by solving certain eikonal and transportequations with the aid of complex rays. The theory is then appliedto several examples: axial waves on a circular cylinder, sphericallysymmetric waves on a sphere, waves on a circular cone with avariable impedance, and waves on the plane boundary of an inhomogeneousmedium. In each case it is found that the asymptotic expansionof the exact solution agrees with the asymptotic solution.     

Arrival times for the wave equation

We establish a definition of arrival time of a wavefront for a propagating wave in anisotropic media that is initially at rest and where the governing partial differential equation is the anisotropic wave equation. This definition of arrival time is not the same as the one in [8, 12]; it eliminates pathological discontinuities that can occur with the older definition and is still consistent with physical intuition. What is substantively new here is that we show that the newly defined arrival time is locally Lipschitz‐continuous. Then following the method in [8, 12] we establish that it satisfies the eikonal equation. Furthermore, in the isotropic case we establish that the arrival time, as defined here, is the unique viscosity solution of the eikonal equation. Our motivation for this work is to use this arrival time at points in the interior of a physical or biological material, which is estimated from displacement measurements, to determine properties of the medium that are represented as functions in the eikonal equation; see [8, 10, 11, 12]. © 2010 Wiley Periodicals, Inc.     

Asymptotic behavior of the wave function of a system of several particles with pair interactions increasing at infinity

We construct an asymptotic representation of the wave functions of systems of two and three quantum particles with pair interactions increasing at infinity. We consider three-particle systems on the line and in the three-dimensional space. The eikonal and transport equations used to construct the asymptotic representation differ significantly from the corresponding equations in the case of decreasing potentials. We study the solution of the nonlinear eikonal equation in detail.     

Global stability for a multi-group SIRS epidemic model with varying population sizes

In this paper, by extending well-known Lyapunov function techniques to SIRS epidemic models, we establish sufficient conditions for the global stability of an endemic equilibrium of a multi-group SIRS epidemic model with varying population sizes which has cross patch infection between different groups. Our proof no longer needs such a grouping technique by graph theory commonly used to analyze the multi-group SIR models.     

An Optimal Control Problem in Coefficients for a Strongly Degenerate Parabolic Equation with Interior Degeneracy

We deal with an optimal control problem in coefficients for a strongly degenerate diffusion equation with interior degeneracy, which is due to the nonnegative diffusion coefficient vanishing with some rate at an interior point of a multi-dimensional space domain. The optimal controller is searched in the class of functions having essentially bounded partial derivatives. The existence of the state system and of the optimal control are proved in a functional framework constructed on weighted spaces. By an approximating control process, explicit approximating optimality conditions are deduced, and a representation theorem allows one to express the approximating optimal control as the solution to the eikonal equation. Under certain hypotheses, further properties of the approximating optimal control are proved, including uniqueness in some situations. The uniform convergence of a sequence of approximating controllers to the solution of the exact control problem is provided. The optimal controller is numerically constructed in a square domain.     

Asymptotic behavior of the solutions of the Helmholtz equation in the caustic shadow domain. I

One makes use of the complex ray method in order to construct the uniform asymptotics of the wave field in the shadow zone beyond the caustic for the Helmholtz equation with an analytic refraction index. The complex eikonal is obtained as a result of the analytic continuation of the eikonal equation into the two-dimensional complex coordinate space. One considers a special example.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 128, pp. 172–185, 1983.The author expresses his gratitude to V. S. Buldyrev for the discussion of the results.     

The Green formula in the theory of potential scattering and corrections to the eikonal scattering amplitude

A method based on the Green formula is developed for calculating the scattering amplitude of fast charged particles in an external field. The scattering amplitude is representable as an integral over an arbitrary closed surface enveloping the domain of influence of the external field on the particle. Corrections to the eikonal scattering amplitude are simply derived without using the specific form of the potential. The resulting formulas can be used to investigate the interaction between particles and fields of complex configuration. Translated from Teoreticheskaya i Matematicheskaya Fizika. Vol. No. 2, pp. 280–288, May, 1998.     

THE ONE-DIMENSIONAL HUGHES MODEL FOR PEDESTRIAN FLOW: RIEMANN-TYPE SOLUTIONS

This paper deals with a coupled system consisting of a scalar conservation law and an eikonal equation,called the Hughes model.Introduced in [24],this model attempts to describe the motion of pedestria...     

On the System of Hamilton–Jacobi and Transport Equations Arising in Geometrical Optics

Abstract

In the present article, we study the system of eikonal and transport equations arising in geometrical optics. The mathematical analysis is performed by using the suitable notion of solution, i.e., the viscosity solution for the Hamilton–Jacobi equation and the measure solution for the transport equation defined via the generalized Filippov characteristics. We study the stability as well as the geometry of the solution to the system.     

On entropy weak solutions of Hughes’ model for pedestrian motion

We consider a generalized version of Hughes’ macroscopic model for crowd motion in the one-dimensional case. It consists in a scalar conservation law accounting for the conservation of the number of pedestrians, coupled with an eikonal equation giving the direction of the flux depending on pedestrian density. As a result of this non-trivial coupling, we have to deal with a conservation law with space–time discontinuous flux, whose discontinuity depends non-locally on the density itself. We propose a definition of entropy weak solution, which allows us to recover a maximum principle. Moreover, we study the structure of the solutions to Riemann-type problems, and we construct them explicitly for small times, depending on the choice of the running cost in the eikonal equation. In particular, aiming at the optimization of the evacuation time, we propose a strategy that is optimal in the case of high densities. All results are illustrated by numerical simulations.