Outgoing Solution Research Articles

High order surface radiation conditions for time-harmonic waves in exterior domains

We formulate a new family of high order on-surface radiation conditions to approximate the outgoing solution to the Helmholtz equation in exterior domains. Motivated by the pseudo-differential expansion of the Dirichlet-to-Neumann operator developed by Antoine et al. (1999), we design a systematic procedure to apply pseudo-differential symbols of arbitrarily high order. Numerical results are presented to illustrate the performance of the proposed method for solving both the Dirichlet and the Neumann boundary value problems. Possible improvements and extensions are also discussed.

Large Outgoing Solutions to Supercritical Wave Equations

We prove the existence of global solutions to the energy-supercritical wave equation in R^{3+1} u_{tt}-\Delta u + |u|^N u = 0, u(0) = u_0, u_t(0) = u_1, 4<N<\infty, for a large class of radially symmetric finite-energy initial data. Functions in this class are characterized as being under the linear flow --- for a specific meaning of outgoing defined below. In particular, we construct global solutions for initial data with large (even infinite) critical Sobolev, Besov, Lebesgue, and Lorentz norms and several other large critical norms.

On the use of Perfectly Matched Layers at corners for scattering problems with sign-changing coefficients

We investigate in a 2D setting the scattering of time-harmonic electromagnetic waves by a plasmonic device, represented as a non-dissipative bounded and penetrable obstacle with a negative permittivity. Using the T-coercivity approach, we first prove that the problem is well-posed in the classical framework Hloc1 if the negative permittivity does not lie in some critical interval whose definition depends on the shape of the device. When the latter has corners, for values inside the critical interval, unusual strong singularities for the electromagnetic field can appear. In that case, well-posedness is obtained by imposing a radiation condition at the corners to select the outgoing black-hole plasmonic wave, that is the one which carries energy towards the corners. A simple and systematic criterion is given to define what is the outgoing solution. Finally, we propose an original numerical method based on the use of Perfectly Matched Layers at the corners. We emphasize that it is necessary to design an ad hoc technique because the field is too singular to be captured with standard finite element methods.

Classification of Hidden Dynamics in Discontinuous Dynamical Systems

Ordinary differential equations with discontinuous right-hand side, where the discontinuity of the vector field arises on smooth surfaces of the phase space, are the topic of this work. The main emphasis is the study of solutions close to the intersection of two discontinuity surfaces. There, the so-called hidden dynamics describe the smooth transition from ingoing to outgoing solution directions, which occurs instantaneously in the jump discontinuity of the vector field. This paper presents a complete classification of such transitions (assuming the vector fields surrounding the intersection are transversal to it). Since the hidden dynamics are realized by standard space regularizations, much insight is obtained for them. One can predict, in the case of multiple solutions of the discontinuous problem, which solution (classical or sliding mode) will be approximated after entering the intersection of two discontinuity surfaces. A novel modification of space regularizations is presented that permits us to avoid (unphysical) high oscillations and makes a numerical treatment more efficient.

Gaussian Beam Methods for the Helmholtz Equation

In this work we construct Gaussian beam approximations to solutions of the high frequency Helmholtz equation with a localized source. Under the assumption of nontrapping rays we show error estimates between the exact outgoing solution and Gaussian beams in terms of the wave number k, both for single beams and superposition of beams. The main result is that the relative local L 2 error in the beam approximations decay as k −N/2 independent of dimension and presence of caustics for N th order beams.

Radiation condition at infinity for the high-frequency Helmholtz equation: optimality of a non-refocusing criterion

We consider the high frequency Helmholtz equation with a variable refraction index $n^2(x)$ ($x \in \R^d$), supplemented with a given high frequency source term supported near the origin $x=0$. A small absorption parameter $\alpha_{\varepsilon}>0$ is added, which somehow prescribes a radiation condition at infinity for the considered Helmholtz equation. The semi-classical parameter is $\varepsilon>0$. We let $\eps$ and $\a_\eps$ go to zero {\em simultaneaously}. We study the question whether the indirectly prescribed radiation condition at infinity is satisfied {\em uniformly} along the asymptotic process $\eps \to 0$, or, in other words, whether the conveniently rescaled solution to the considered equation goes to the {\em outgoing} solution to the natural limiting Helmholtz equation. This question has been previously studied by the first autor. It is proved that the radiation condition is indeed satisfied uniformly in $\eps$, provided the refraction index satisfies a specific {\em non-refocusing condition}, a condition that is first pointed out in this reference. The non-refocusing condition requires, in essence, that the rays of geometric optics naturally associated with the high-frequency Helmholtz operator, and that are sent from the origin $x=0$ at time $t=0$, should not refocus at some later time $t>0$ near the origin again. In the present text we show the {\em optimality} of the above mentionned non-refocusing condition, in the following sense. We exhibit a refraction index which {\em does} refocus the rays of geometric optics sent from the origin near the origin again, and, on the other hand, we completely compute the asymptotic behaviour of the solution to the associated Helmholtz equation: we show that the limiting solution {\em does not} satisfy the natural radiation condition at infinity. More precisely, we show that the limiting solution is a {\em perturbation} of the outgoing solution to the natural limiting Helmholtz equation, and that the perturbing term explicitly involves the contribution of the rays radiated from the origin which go back to the origin. This term is also conveniently modulated by a phase factor, which turns out to be the action along the above rays of the hamiltonian associated with the semiclassical Helmholtz equation.

Limiting Absorption Principle and Perfectly Matched Layer Method for Dirichlet Laplacians in Quasi-cylindrical Domains

We establish a limiting absorption principle for Dirichlet Laplacians in quasi-cylindrical domains. Outside a bounded set these domains can be transformed onto a semi-cylinder by suitable diffeomorphisms. Dirichlet Laplacians model quantum or acoustically-soft waveguides associated with quasi-cylindrical domains. We construct a uniquely solvable problem with perfectly matched layers of finite length. We prove that solutions of the latter problem approximate outgoing or incoming solutions with an error that exponentially tends to zero as the length of layers tends to infinity. Outgoing and incoming solutions are characterized by means of the limiting absorption principle.

Perfectly Matched Layers for Diffraction Gratings in Inhomogeneous Media. Stability and Error Estimates

We study the scattering of acoustic or electromagnetic waves at diffraction gratings in inhomogeneous media. The refractive index stabilizes to 1 as the distance to the grating increases. Outgoing solutions are characterized by means of the limiting absorption principle. We prove the unique solvability of the problem with perfectly matched layer of finite length. Further, we show that solutions of the latter problem approximate outgoing solutions of the original problem with an error that exponentially tends to zero as the length of perfectly matched layer tends to infinity. Contribution of eigenvalues and resonances to the error of approximation is clarified.

The Limiting Absorption Principle for a Periodic Semi-Infinite Waveguide

We study an elliptic boundary value problem in a semi-infinite, straight waveguide containing a periodic material distribution. The spectral parameter is taken to lie in a spectral band of the corresponding full-space periodic operator. In contrast to the situation with constant coefficients, it is not obvious how to define the “outgoing” solution of the problem. In order to define the outgoing solution, we introduce a small absorption to regularize the problem and then prove that the solutions converge locally in $H^1$.

Radiation boundary conditions for the numerical solution of the three-dimensional time-dependent Schrödinger equation with a localized interaction

Exact radiation boundary conditions on the surface of a sphere are presented for the single-particle time-dependent Schrödinger equation with a localized interaction. With these boundary conditions, numerical computations of spatially unbounded outgoing wave solutions can be restricted to the finite volume of a sphere. The boundary conditions are expressed in terms of the free-particle Green's function for the outside region. The Green's function is analytically calculated by an expansion in spherical harmonics and by the method of Laplace transformation. For each harmonic number a discrete boundary condition between the function values at adjacent radial grid points is obtained. The numerical method is applied to quantum tunneling through a spherically symmetric potential barrier with different angular-momentum quantum numbers l . Calculations for l=0 are compared to exact theoretical results.

Klein paradox and scattering theory for the semi-classical Dirac equation

We study the Klein paradox for the semi-classical Dirac operator on R with potentials having constant limits, possibly different at infinity. Using the complex WKB method, the time-independent scattering theory in terms of incoming and outgoing plane wave solutions is established. The corresponding scattering matrix is unitary. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in the cases of the Klein paradox, the total transmission and the total reflection. Finally, we treat the scattering problem in the zero mass case.

The Helmholtz Equation in a Locally Perturbed Half-Space with Non-Absorbing Boundary

We obtain uniqueness and existence results of an outgoing solution for the Helmholtz equation in a half-space, or in a compact local perturbation of it, with an impedance boundary condition. It is worth noting that these kinds of domains have unbounded boundaries which lead to a non-classical exterior problem. The established radiation condition is somewhat different from the usual Sommerfeld’s one, due to the appearance of surface waves (in the case of a non-absorbing boundary). A half-space Green’s function framework is used to carry out our computations. This is an extended and detailed version of the previous article “The Helmholtz equation with impedance in a half-space,” Duran et al. (CR Acad Sci Paris Ser I 341:561–566, 2005).

Acoustic propagation in a flow: numerical simulation of the time-harmonic regime

We consider the time-harmonic acoustic radiation of a source in a moving fluid. The problem is set in a two-dimensional infinite duct and the mean flow is a subsonic parallel shear flow, with a regular profile. We deal with an equation (due to Galbrun) whose unknown is the displacement perturbation. We show how to solve the problem with a finite element method by writing a regularized or augmented formulation and using Perfectly Matched Layers to select the outgoing solution. Due to the presence of a non-local term coming from the regularization, an iterative process of resolution is preferred, which converges faster for weaker shear. Some mathematical results are established in the dissipative case. Numerical illustrations are finally presented.

Local fields for asymptotic matching in multidimensional mode conversion

The problem of resonant mode conversion in multiple spatial dimensions is considered. Using phase space methods, a complete theory is developed for constructing matched asymptotic expansions that fit incoming and outgoing WKB solutions. These results provide, for the first time, a complete and practical method for including multidimensional conversion in ray tracing algorithms. The paper provides a self-contained description of the following topics: (1) how to use eikonal (also known as ray tracing or WKB) methods to solve vector wave equations and how to detect conversion regions while following rays; (2) once conversion is detected, how to fit to a generic saddle structure in ray phase space associated with the most common type of conversion; (3) given the saddle structure, how to carry out a local projection of the full vector wave equation onto a local two-component normal form that governs the two resonantly interacting waves. This determines both the uncoupled dispersion functions and the coupling constant, which in turn determine the uncoupled WKB solutions; (4) given the normal form of the local two-component wave equation, how to find the particular solution that matches the amplitude, phase, and polarization of the incoming ray, to the amplitude, phase, and polarization of the two outgoing rays: the transmitted and converted rays.

Perfectly Matched Layers for Time-Harmonic Acoustics in the Presence of a Uniform Flow

This paper is devoted to the resolution of the time-harmonic linearized Galbrun equation, which models, via a mixed Lagrangian--Eulerian representation, the propagation of acoustic and hydrodynamic perturbations in a given flow of a compressible fluid. We consider here the case of a uniform subsonic flow in an infinite, two-dimensional duct. Using a limiting absorption process, we characterize the outgoing solution radiated by a compactly supported source. Then we propose a Fredholm formulation with perfectly matched absorbing layers for approximating this outgoing solution. The convergence of the approximated solution to the exact one is proved, and error estimates with respect to the parameters of the absorbing layers are derived. Several significant numerical examples are included.

Photochemical Reactivity of Trifluoromethyl Aromatic Amines: The Example of 3,5‐diamino‐trifluoromethyl‐benzene (3,5‐DABTF)

This work presents the application of an on-line photoreactor to the detection of 3,5-diamino-trifluoromethyl-benzene (3,5-DABTF) in aqueous solutions. When irradiated at 310 nm, this compound is defluorinated to 3,5-diaminobenzoic acid by a nucleophilic substitution of the fluoride by water. Concomitantly, defluorination intermediates polymerize through amide bonds to give dark-colored compounds. We take advantage of the photocatalyzed defluorination and the subsequent decrease in pH to develop an original and specific photoreactor. Continuous recording of pH and temperature in the outlet of the reactor by a dual electrode gives us an opportunity to optimize the system. In the photoreactor, 3,5-DABTF is immediately and totally transformed as attested by the rapid drop of the flowing solution pH from 6.2 to 3.2 and the chromatographic analysis of the outgoing solutions. The detection remains effective from 1 to 1000 parts per million.

The Helmholtz equation with impedance in a half-space

In this Note we obtain existence and uniqueness results for the Helmholtz equation in the half-space R + 3 with an impedance or Robin boundary condition. Basically, we follow the procedure we have already used in the bi-dimensional case (the half-plane). Thus, we compute the associated Green's function with the help of a double Fourier transform and we analyze its far field in order to obtain radiation conditions that allow us to prove the uniqueness of an outgoing solution. Again, these radiation conditions are somewhat unusual due to the appearance of a surface wave guided by the boundary. An integral representation of the solution is presented by means of the Green's function and the boundary data. To cite this article: M. Durán et al., C. R. Acad. Sci. Paris, Ser. I 341 (2005).

The radiation condition at infinity for the high-frequency Helmholtz equation with source term: a wave-packet approach

We consider the high-frequency Helmholtz equation with a given source term, and a small absorption parameter α > 0 . The high-frequency (or: semi-classical) parameter is ɛ > 0 . We let ɛ and α go to zero simultaneously. We assume that the zero energy is non-trapping for the underlying classical flow. We also assume that the classical trajectories starting from the origin satisfy a transversality condition, a generic assumption. Under these assumptions, we prove that the solution u ɛ radiates in the outgoing direction, uniformly in ɛ . In particular, the function u ɛ , when conveniently rescaled at the scale ɛ close to the origin, is shown to converge towards the outgoing solution of the Helmholtz equation, with coefficients frozen at the origin. This provides a uniform version (in ɛ ) of the limiting absorption principle. Writing the resolvent of the Helmholtz equation as the integral in time of the associated semi-classical Schrödinger propagator, our analysis relies on the following tools: (i) for very large times, we prove and use a uniform version of the Egorov Theorem to estimate the time integral; (ii) for moderate times, we prove a uniform dispersive estimate that relies on a wave-packet approach, together with the above-mentioned transversality condition; (iii) for small times, we prove that the semi-classical Schrödinger operator with variable coefficients has the same dispersive properties as in the constant coefficients case, uniformly in ɛ .

Periodic standing-wave approximation: Overview and three-dimensional scalar models

The periodic standing-wave method for binary inspiral computes the exact numerical solution for periodic binary motion with standing gravitational waves, and uses it as an approximation to slow binary inspiral with outgoing waves. Important features of this method presented here are: (i) the mathematical nature of the ``mixed'' partial differential equations to be solved, (ii) the meaning of standing waves in the method, (iii) computational difficulties, and (iv) the ``effective linearity'' that ultimately justifies the approximation. The method is applied to three-dimensional nonlinear scalar model problems, and the numerical results are used to demonstrate extraction of the outgoing solution from the standing-wave solution, and the role of effective linearity.

Rapid evaluation of radiation boundary kernels for time-domain wave propagation on blackholes: theory and numerical methods

For scalar, electromagnetic, or gravitational wave propagation on a fixed Schwarzschild blackhole background, we describe the exact nonlocal radiation outer boundary conditions (ROBC) appropriate for a spherical outer boundary of finite radius enclosing the blackhole. Derivation of the ROBC is based on Laplace and spherical-harmonic transformation of the Regge–Wheeler equation, the PDE governing the wave propagation, with the resulting radial ODE an incarnation of the confluent Heun equation. For a given angular index l the ROBC feature integral convolution between a time-domain radiation boundary kernel (TDRK) and each of the corresponding 2 l+1 spherical-harmonic modes of the radiation field. The TDRK is the inverse Laplace transform of a frequency-domain radiation kernel (FDRK) which is essentially the logarithmic derivative of the asymptotically outgoing solution to the radial ODE. We develop several numerical methods for examining the frequency dependence of both the outgoing solution and the FDRK. Using these methods we numerically implement the ROBC in a follow-up article. Our work is a partial generalization to Schwarzschild wave propagation and Heun functions of the methods developed for flatspace wave propagation and Bessel functions by Alpert, Greengard, and Hagstrom (AGH), save for one key difference. Whereas AGH had the usual armamentarium of analytical results (asymptotics, order recursion relations, bispectrality) for Bessel functions at their disposal, what we need to know about Heun functions must be gathered numerically as relatively less is known about them.