The uniqueness and existence of solutions for the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation

In this paper, we study the 3D Helmholtz equation in a step‐index waveguide with unbounded perturbation, allowing the presence of guided waves. Our assumptions on the perturbed and source terms are too few. On the basis of the Green's function for the 3D homogeneous Helmholtz equation in a step‐index waveguide without perturbation, we introduce a generalized (out‐going) Sommerfeld–Rellich radiation condition, and then we prove the uniqueness and existence of solutions for the studied 3D Helmholtz equation satisfying our radiation condition. Copyright © 2012 John Wiley & Sons, Ltd.     

Unique continuation on a line for the Helmholtz equation

In this article, local unique continuation on a line for solutions of the Helmholtz equation is discussed. The fundamental solution of the exterior problem for the Helmholtz equation have a logarithmic singularity which behaves similar to those of the interior problem for the Laplace equation in two dimension. A Hölder-type conditional stability estimate of the proposed exterior problem for the Helmholtz equation is obtained by adopting the complex extension method in Cheng and Yamamoto [J. Cheng and M. Yamamoto, Unique continuation on a line for harmonic functions, Inverse Probl. 14 (1998), pp. 869–882]. Finally, a regularization scheme based on the collocation method is compatible with the Hölder-type stability estimate provided that the line does not intersect the boundary of the domain for both the Laplace and the Helmholtz equations.     

A fourth-order Magnus scheme for Helmholtz equation

For wave propagation in a slowly varying waveguide, it is necessary to solve the Helmholtz equation in a domain that is much larger than the typical wavelength. Standard finite difference and finite element methods must resolve the small oscillatory behavior of the wave field and are prohibitively expensive for practical applications. A popular method is to approximate the waveguide by segments that are uniform in the propagation direction and use separation of variables in each segment. For a slowly varying waveguide, it is possible that the length of such a segment is much larger than the typical wavelength. To reduce memory requirements, it is advantageous to reformulate the boundary value problem of the Helmholtz equation as an initial value problem using a pair of operators. Such an operator-marching scheme can also be solved with the piecewise uniform approximation of the waveguide. This is related to the second-order midpoint exponential method for a system of linear ODEs. In this paper, we develop a fourth-order operator-marching scheme for the Helmholtz equation using a fourth-order Magnus method.     

A multigrid solver to the Helmholtz equation with a point source based on travel time and amplitude

The Helmholtz equation arises when modeling wave propagation in the frequency domain. The equation is discretized as an indefinite linear system, which is difficult to solve at high wave numbers. In many applications, the solution of the Helmholtz equation is required for a point source. In this case, it is possible to reformulate the equation as two separate equations: one for the travel time of the wave and one for its amplitude. The travel time is obtained by a solution of the factored eikonal equation, and the amplitude is obtained by solving a complex‐valued advection–diffusion–reaction equation. The reformulated equation is equivalent to the original Helmholtz equation, and the differences between the numerical solutions of these equations arise only from discretization errors. We develop an efficient multigrid solver for obtaining the amplitude given the travel time, which can be efficiently computed. This approach is advantageous because the amplitude is typically smooth in this case and, hence, more suitable for multigrid solvers than the standard Helmholtz discretization. We demonstrate that our second‐order advection–diffusion–reaction discretization is more accurate than the standard second‐order discretization at high wave numbers, as long as there are no reflections or caustics. Moreover, we show that using our approach, the problem can be solved more efficiently than using the common shifted Laplacian multigrid approach.     

Sommerfeld condition for a Liouville equation and concentration of trajectories

We analyse the concentration of trajectories in a Liouville equation set in the full space with a potential which is not constant at infinity. Our motivation comes from geometrical optics where it appears as the high freqency limit of Helmholtz equation. We conjecture that the mass and energy concentrate on local maxima of the refraction index and prove a result in this direction. To do so, we establish a priori estimates in appropriate weighted spaces and various forms of a Sommerfeld radiation condition for solutions of such a stationary Liouville equation.Dedicated to IMPA on the occasion of its 50^th anniversary     

A FOURTH ORDER DERIVATIVE-FREE OPERATOR MARCHING METHOD FOR HELMHOLTZ EQUATION IN WAVEGUIDES

A fourth-order operator marching method for the Helmholtz equation in a waveguide is developed in this paper. It is derived from a new fourth-order exponential integrator for linear evolution equations. The method improves the second-order accuracy associated with the widely used step-wise coupled mode method where the waveguide is approximated by segments that are uniform in the propagation direction. The Helmholtz equation is solved using a one-way reformulation based on the Dirichlet-to-Neumann map. An alternative version closely related to the coupled mode method is also given. Numerical results clearly indicate that the method is more accurate than the coupled mode method while the required computing effort is nearly the same.     

High-frequency limit of the Helmholtz equation with variable refraction index

We study the high-frequency limit of the Helmholtz equation with variable refraction index and a source term concentrated near a p-dimensional affine subspace. Under some conditions, we first derive uniform estimates in Besov spaces for the solutions. Then, we prove that the semi-classical measure associated with these solutions satisfies the stationary Liouville equation with an explicit source term and has certain radiation property at infinity.     

A sixth-order dual preserving algorithm for the Camassa-Holm equation

The paper presents a sixth-order numerical algorithm for studying the completely integrable Camassa-Holm (CH) equation. The proposed sixth-order accurate method preserves both the dispersion relation and the Hamiltonians of the CH equation. The CH equation in this study is written as an evolution equation, involving only the first-order spatial derivatives, coupled with the Helmholtz equation. We propose a two-step method that first solves the evolution equation by a sixth-order symplectic Runge-Kutta method and then solves the Helmholtz equation using a three-point sixth-order compact scheme. The first-order derivative terms in the first step are approximated by a sixth-order dispersion-relation-preserving scheme that preserves the physically inherent dispersive nature. The compact Helmholtz solver, on the other hand, allows us to use relatively few nodal points in a stencil, while achieving a higher-order accuracy. The sixth-order symplectic Runge-Kutta time integrator is preferable for an equation that possesses a Hamiltonian structure. We illustrate the ability of the proposed scheme by examining examples involving peakon or peakon-like solutions. We compare the computed solutions with exact solutions or asymptotic predictions. We also demonstrate the ability of the symplectic time integrator to preserve the Hamiltonians. Finally, via a smooth travelling wave problem, we compare the accuracy, elapsed computing time, and rate of convergence among the proposed method, a second-order two-step algorithm, and a completely integrable particle method.     

On Vector Helmholtz Equation with a Coupling Boundary Condition

The Helmholtz equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown.In this paper, we study the vector Helmholtz problem in domains of both C~(1,1)and Lipschitz.We es- tablish a rigorous variational analysis such as equivalence,existence and uniqueness. And we propose finite element approximations based on the uncoupled solutions.Fi- nally we present a convergence analysis and error estimates.     

Normalized system of functions with respect to the Laplace operator and its applications

A system of functions 0-normalized with respect to the operator Δ in some domain is constructed. Application of this system to boundary value problems for the polyharmonic equation is considered. Connection between harmonic functions and solutions of the Helmholtz equation is investigated.     

On a class of preconditioners for solving the Helmholtz equation

In 1983, a preconditioner was proposed [J. Comput. Phys. 49 (1983) 443] based on the Laplace operator for solving the discrete Helmholtz equation efficiently with CGNR. The preconditioner is especially effective for low wavenumber cases where the linear system is slightly indefinite. Laird [Preconditioned iterative solution of the 2D Helmholtz equation, First Year's Report, St. Hugh's College, Oxford, 2001] proposed a preconditioner where an extra term is added to the Laplace operator. This term is similar to the zeroth order term in the Helmholtz equation but with reversed sign. In this paper, both approaches are further generalized to a new class of preconditioners, the so-called “shifted Laplace” preconditioners of the form Δφ−k²φ with . Numerical experiments for various wavenumbers indicate the effectiveness of the preconditioner. The preconditioner is evaluated in combination with GMRES, Bi-CGSTAB, and CGNR.     

Computation of waveguide scattering matrix near thresholds

A waveguide occupies infinite strip with one or several narrows on a two-dimensional (2D) plane and is governed by the Helmholtz equation with Dirichlet boundary condition. On the waveguide continuous spectrum, which coincides with a half-axis, a scattering matrix is defined. At each point of the continuous spectrum this matrix has finite size, which changes at thresholds. The thresholds form a sequence of positive numbers increasing to infinity. Approximate calculation of the scattering matrix in a threshold vicinity requires special treatment. We discuss and compare two methods of numerical approximation to the scattering matrix near a threshold.     

A radiation condition for uniqueness in a wave propagation problem for 2‐D open waveguides

We study the uniqueness of solutions of Helmholtz equation for a problem that concerns wave propagation in waveguides. The classical radiation condition does not apply to our problem because the inhomogeneity of the index of refraction extends to infinity in one direction. Also, because of the presence of a waveguide, some waves propagate in one direction with different propagation constants and without decaying in amplitude. We provide an explicit condition for uniqueness for rectilinear waveguides, which takes into account the physically significant components, corresponding to guided and non‐guided waves; this condition reduces to the classical Sommerfeld–Rellich condition in the relevant cases. By a careful asymptotic analysis we prove that the solution derived by Magnanini and Santosa (SIAM J. Appl. Math. 2001; 61 :1237–1252) for stratified media satisfies our radiation condition. Copyright © 2008 John Wiley & Sons, Ltd.     

Application of Adapted-Bubbles to the Helmholtz Equation with Large Wavenumbers in 2D

An adapted-bubbles approach which is a modification of the residual-free bubbles (RFB) method, is proposed for the Helmholtz problem in 2D. A new two-level finite element method is introduced for the approximations of the bubble functions. Unlike the other equations such as the advection-diffusion equation, RFB method when applied to the Helmholtz equation, does not depend on another stabilized method to obtain approximations to the solutions of the sub-problems. Adapted-bubbles (AB) are obtained by a simple modification of the sub-problems. This modification increases the accuracy of the numerical solution impressively. We provide numerical experiments with the AB method up to $ch = 5$ where $c$ is the wavenumber and $h$ is the mesh size. Numerical tests show that the AB method is better by far than higher order methods available in the literature.     

The Helmholtz equation in a locally perturbed half-plane with passive boundary

^** Email: mduran{at}ing.puc.cl^*** Email: ignacio.muga{at}ucv.cl^**** Email: nedelec{at}cmapx.polytechnique.fr In this article, we study the existence and uniqueness of outgoingsolutions for the Helmholtz equation in locally perturbed half-planeswith passive boundary. We establish an explicit outgoing radiationcondition which is somewhat different from the usual Sommerfeld'sone due to the appearance of surface waves. We work with thehelp of Fourier analysis and a half-plane Green's function framework.This is an extended and detailed version of the previous articleDurán et al. (2005, The Helmholtz equation with impedancein a half-plane. C. R. Acad. Sci. Paris, Ser. I, 340, 483–488).     

On a quasi-reversibility regularization method for a Cauchy problem of the Helmholtz equation

In this paper, we consider the Cauchy problem for the Helmholtz equation in a rectangle, where the Cauchy data is given for y=0 and boundary data are for x=0 and x=π. The solution is sought in the interval 0<y≤1. A quasi-reversibility method is applied to formulate regularized solutions which are stably convergent to the exact one with explicit error estimates.     

THE NUMERICAL SOLUTION OF FIRST KIND INTEGRAL EQUATION FOR THE HELMHOLTZ EQUATION ON SMOOTH OPEN ARCS

1. IntroductiouThe mathewtical tratod of the scattering Of theharmonic acoustic or electromagnoticwaves by an Mtely lOng sethecylindrical obstacle with a 8mooth opeu coDtour crewSeCtboF C Rs Ieads to unbounded boundare wtue problems for the Helmhltz equabo I3lwith wave nUmer h > 0.In the singtelayer Woach one Seeks the solutbo in the formwhere d8. is the element of arc length, and the fundamental solUbo to the Helmholtz equatfonis giveu byin terms Of the Hds fUnction H6') of order zero…     

An expansion theorem for two‐dimensional elastic waves and its application

We prove an Atkinson–Wilcox‐type expansion for two‐dimensional elastic waves in this paper. The approach developed on the two‐dimensional Helmholtz equation will be applied in the proof. When the elastic fields are involved, the situation becomes much harder due to two wave solutions propagating at different phase velocities. In the last section, we give an application about the reconstruction of an obstacle from the scattering amplitude. Copyright © 2006 John Wiley & Sons, Ltd.     

On reflection principles supported on a finite set

We investigate the existence of reflection formulas supported on a finite set. It is found that for solutions of the Laplace and Helmholtz equation there are no finitely supported reflection principles unless the support is a single point. This confirms that in order to construct a reflection formula that is not ‘point to point’, it is necessary to consider a continuous support. For solutions of the wave equation ^∂2u/∂x∂y=0, there exist finitely supported reflection principles that can be constructed explicitly. For solutions of the telegraph equation ^∂2u/∂x∂y+λ²u=0, we show that if a reflection principle is supported on less than five points then it is a point to point reflection principle.     

Multi‐periodic eigensolutions to the Dirac operator and applications to the generalized Helmholtz equation on flat cylinders and on the n‐torus

In this paper, we study the solutions to the generalized Helmholtz equation with complex parameter on some conformally flat cylinders and on the n‐torus. Using the Clifford algebra calculus, the solutions can be expressed as multi‐periodic eigensolutions to the Dirac operator associated with a complex parameter λ∈?. Physically, these can be interpreted as the solutions to the time‐harmonic Maxwell equations on these manifolds. We study their fundamental properties and give an explicit representation theorem of all these solutions and develop some integral representation formulas. In particular, we set up Green‐type formulas for the cylindrical and toroidal Helmholtz operator. As a concrete application, we explicitly solve the Dirichlet problem for the cylindrical Helmholtz operator on the half cylinder. Finally, we introduce hypercomplex integral operators on these manifolds, which allow us to represent the solutions to the inhomogeneous Helmholtz equation with given boundary data on cylinders and on the n‐torus. Copyright © 2009 John Wiley & Sons, Ltd.