Wave Equation In Domain Research Articles

Energy decay for linear dissipative wave equations in exterior domains

In earlier works [2,12], we have shown the uniform decay of the local energy of the damped wave equation in exterior domain when the damper is spatially localized near captive rays. In order to have uniform decay of the total energy, the damper has also to act at space infinity. In this work, we establish uniform decay of both the local and global energies. The rates of decay turn out to be the same as those for the heat equation, which shows that an effective damper at space infinity strengthens the parabolic structure in the equation.

Generalized convolution quadrature with variable time stepping. Part II: Algorithm and numerical results

In this paper we address the implementation of the Generalized Convolution Quadrature (gCQ) presented and analyzed by the authors in a previous paper for solving linear parabolic and hyperbolic convolution equations. Our main goal is to overcome the current restriction to uniform time steps of Lubich's Convolution Quadrature (CQ). A major challenge for the efficient realization of the new method is the evaluation of high-order divided differences for the transfer function in a fast and stable way. Our algorithm is based on contour integral representation of the numerical solution and quadrature in the complex plane. As the main application we consider the wave equation in exterior domains, which is formulated as a retarded boundary integral equation. We provide numerical experiments to illustrate the theoretical results.

Numerical investigation of vibration and dynamic pressure of a vertical axis wind turbine

In the environmental field, the problems of noise reduction have become a major preoccupation, particularly on the noise generated by the acoustic radiation pressure produced by wind turbines. This paper is aimed at presenting the investigation on the application of variational indirect boundary element method for study the acoustic radiation pressure produced by vertical-axis wind turbine. For this initiative, we considered Neumann boundary condition. The formulation has two advantages: the first one is to avoid the meshing of the fluid domain; the second advantage is to treat the singular integral of the Green's function, solution of fundamental solution of the wave equation in frequency domain.

Convergence of infinite element methods for scalar waveguide problems

We consider the numerical solution of scalar wave equations in domains which are the union of a bounded domain and a finite number of infinite cylindrical waveguides. The aim of this paper is to provide a new convergence analysis of both the Perfectly Matched Layer (PML) method and the Hardy space infinite element method in a unified framework. We treat both diffraction and resonance problems. The theoretical error bounds are compared with errors in numerical experiments.

Feasible fundamental solution of the multiphysics wave equation in inhomogeneous domains of complex shape

Fundamental solutions of the linear equations governing mechanical and electromagnetic oscillations are kinematically represented by delay time along ray trajectories. The fundamental solutions can contain components which are not physically justified, if their ray trajectories are partly located outside the actual medium in accordance with Fermat’s principle. To exclude all non-physical components and consider only the physically feasible fundamental solution, ray trajectories and delay time must satisfy the generalized Fermat’s principle, as introduced by Hadamard in 1910. We introduce a rigorous dynamic description of this feasible fundamental solution satisfying the generalized Fermat’s principle and being physically justifiable. The description is based on an integral condition of absolute absorption at the boundary of an effective medium. This condition selects a subset of the physically feasible fundamental solutions. We prove that, in homogeneous domains, the feasible fundamental solution is the sum of Green’s function for unbounded medium and an operator Neumann series describing cascade diffraction at the boundary. In inhomogeneous domains we represent the feasible fundamental solution by an equation with a volume integral operator. The integral kernel contains the feasible fundamental solution for a homogeneous domain. We introduce feasible surface and volume integral operators that eliminate the unfeasible wavefields in the geometrical shadow zones.

Tri-prong scheme for regularized long wave equation

This paper witnesses the application of a tri-prong scheme comprising the well-known Variational Iteration (VIM), Adomian’s polynomials and an auxiliary parameter to obtain solutions of regularized long wave (RLW) equation in large domain. Computational work elucidates the solution procedure appropriately and comparison with results by the standard variational iteration method shows that the auxiliary parameter proves very effective to control the convergence region of approximate solutions.

Super-Grid Modeling of the Elastic Wave Equation in Semi-Bounded Domains

AbstractWe develop a super-grid modeling technique for solving the elastic wave equation in semi-bounded two- and three-dimensional spatial domains. In this method, waves are slowed down and dissipated in sponge layers near the far-field boundaries. Mathematically, this is equivalent to a coordinate mapping that transforms a very large physical domain to a significantly smaller computational domain, where the elastic wave equation is solved numerically on a regular grid. To damp out waves that become poorly resolved because of the coordinate mapping, a high order artificial dissipation operator is added in layers near the boundaries of the computational domain. We prove by energy estimates that the super-grid modeling leads to a stable numerical method with decreasing energy, which is valid for heterogeneous material properties and a free surface boundary condition on one side of the domain. Our spatial discretization is based on a fourth order accurate finite difference method, which satisfies the principle of summation by parts. We show that the discrete energy estimate holds also when a centered finite difference stencil is combined with homogeneous Dirichlet conditions at several ghost points outside of the far-field boundaries. Therefore, the coefficients in the finite difference stencils need only be boundary modified near the free surface. This allows for improved computational efficiency and significant simplifications of the implementation of the proposed method in multi-dimensional domains. Numerical experiments in three space dimensions show that the modeling error from truncating the domain can be made very small by choosing a sufficiently wide super-grid damping layer. The numerical accuracy is first evaluated against analytical solutions of Lamb’s problem, where fourth order accuracy is observed with a sixth order artificial dissipation. We then use successive grid refinements to study the numerical accuracy in the more complicated motion due to a point moment tensor source in a regularized layered material.

An efficient wavenumber–space–time domain finite-difference modeling of acoustic wave equation for synthesizing CMP gathers

The wavenumber–space–time domain wave equation is derived from the 2D time-domain wave equation in the horizontally stratified media by taking the Fourier transform of the equation with respect to one of the spatial variables. The derived wave equation is solved by the finite-difference method and analyzed in terms of stability and dispersion. Although there are lots of wavenumbers to be solved according to the sampling theorem, the modeling can be accelerated by limiting the range of wavenumber used for the inverse Fourier transform. The proposed method generates common mid-point (CMP) gathers directly without sorting process by applying the shifting theorem. We show that our algorithm is efficient and generates the same results as ones from the 2D modeling through two numerical examples. The proposed modeling algorithm can be used for the inversion of CMP gathers or amplitude-versus-offset inversion.

Almost global existence for 4-dimensional quasilinear wave equations in exterior domains

This article focuses on almost global existence for quasilinear wave equations with small initial data in 4-dimensional exterior domains. The nonlinearity is allowed to depend on the solution at the quadratic level as well as its first and second derivatives. In the boundaryless setting, Hörmander proved that the lifespan $T_\epsilon \gtrsim \exp(c/\epsilon)$, where $\epsilon\gt 0$ denotes the size of the Cauchy data. Later Du, the second author, Sogge, and Zhou showed that this inequality also holds for star-shaped obstacles. Following up on the authors' work in the 3-dimensional case, we only require that the obstacle allow for a sufficiently rapid decay of local energy for the linear homogeneous wave equation. The key innovation of this paper is the combination of the boundary term estimates of the second author and Sogge with a variant of an estimate of Klainerman and Sideris, which is obtained via a Sobolev inequality of Du and Zhou.

Frequency domain optoacoustic tomography using amplitude and phase

We introduce optoacoustic tomographic imaging using intensity modulated light sources and collecting amplitude and phase information in the frequency domain. Imaging is performed at multiple modulation frequencies. The forward modeling uses the Green's function solution to the pressure wave equation in frequency domain and the resulting inverse problem is solved using regularized least squares minimization. We study the effect of the number of frequencies and of the bandwidth employed on the image quality achieved. The possibility of employing an all-frequency domain optoacoustic imaging for experimental measurements is studied as a function of noise. We conclude that frequency domain optoacoustic tomography may evolve to a practical experimental method using light intensity modulated sources, with advantages over time-domain optoacoustics.

Angle-Domain Gathers Computation Using Poynting Vector of Two-Way Acoustic Wave Equation

Based on two-way acoustic wave equation, we present a method for computing angle-domain common-image gathers for reverse time migration. The method calculates the propagation direction of source wave-fields and receiver wave-fields according to expression of energy flow density vectors (Poynting vectors) of acoustic wave equation in space-time domain to obtain the reflection angle, then apply the normalized cross-correlation imaging condition to achieve the angle-domain common-image gathers. The angle gathers obtained can be used for migration velocity analysis, AVA analysis and so on. Numerical examples and real data examples demonstrate the effectiveness of this method.

Exact controllability for a wave equation with fixed boundary control

This paper addresses the study of the controllability for a one-dimensional wave equation in domains with moving boundary. This equation characterizes the motion of a string with a fixed endpoint and the other one moving. When the speed of the moving endpoint is less than , by the Hilbert Uniqueness Method, the exact controllability of this equation is established. Also, the explicit dependence of the controllability time on the speed of the moving endpoint is given.

Multiscale asymptotic expansions methods and numerical algorithms for the wave equations in perforated domains

In this paper, we are concerned with the wave equations in perforated domains with a homogeneous Neumann condition on the boundary of the holes. The multiscale asymptotic expansion of the solution for the problem are constructed and associated explicit convergence rates are obtained. A multiscale numerical method is introduced. Finally, we present some numerical results which support strongly the convergence theorem.

The effect of fast and continuous change of dielectric permittivity on whispering gallery modes in circular microcavity

The detailed analysis of the response of whispering gallery mode (WGM) to the permittivity change in circular microcavities made of materials with noteworthy relaxation time is presented. Continuous and mathematically well behavior function is employed to describe temporal change of resonator permittivity and the wave equation in both frequency and time domains is analyzed by means of inverse Laplace transform method. The exact transient response of cavity resonance mode with the steady state solutions are described carefully and it is shown that fast and continuous change of resonator dielectric permittivity behave like several propagating boundaries inside and outside the cavity. The transformed WGM appears after transient duration. These analyses are providing suitable tools for investigation of dynamic optical devices.

Controllability for a Wave Equation with Moving Boundary

We investigate the controllability for a one-dimensional wave equation in domains with moving boundary. This model characterizes small vibrations of a stretched elastic string when one of the two endpoints varies. When the speed of the moving endpoint is less than1-1/e, by Hilbert uniqueness method, sidewise energy estimates method, and multiplier method, we get partial Dirichlet boundary controllability. Moreover, we will give a sharper estimate on controllability time that only depends on the speed of the moving endpoint.

Decay estimates of solutions for mildly degenerate Kirchhoff type dissipative wave equations in unbounded domains

Decay estimates of solutions for mildly degenerate Kirchhoff type dissipative wave equations in unbounded domains

ASYMPTOTICS OF THE CRITICAL NONLINEAR WAVE EQUATION FOR A CLASS OF NON-STAR-SHAPED OBSTACLES

Scattering for the energy critical nonlinear wave equation for domains exterior to non-trapping obstacles in 3+1 dimension is known for the star-shaped case. In this paper, we extend the scattering for a class of non-star-shaped obstacles called illuminated from exterior. The main tool we use is the method of multipliers with weights that generalize the Morawetz multiplier to suit the geometry of the obstacle.

Retarded boundary integral equations on the sphere: exact and numerical solution

In this paper we consider the three-dimensional wave equation in unbounded domains with Dirichlet boundary conditions. We start from a retarded single layer potential ansatz for the solution of these equations which leads to the retarded potential integral equation (RPIE) on the bounded surface of the scatterer. We formulate an algorithm for the spacetime Galerkin discretization with smooth and compactly supported temporal basis functions which have been introduced in [S. Sauter and A. Veit: A Galerkin Method for Retarded Boundary Integral Equations with Smooth and Compactly Supported Temporal Basis Functions, Preprint 04-2011, Universitat Zurich]. For the debugging of an implementation and for systematic parameter tests it is essential to have some explicit representations and some analytic properties of the exact solutions for some special cases at hand. We will derive such explicit representations for the case that the scatterer is the unit ball. The obtained formulas are easy to implement and we will present some numerical experiments for these cases to illustrate the convergence behaviour of the proposed method. AMS subject classifications. 35L05, 65R20

On a mixed problem with a nonlinear acoustic boundary condition for a non-locally reacting boundaries

We discuss global solvability to the mixed problem for a nonlinear wave equation in domains with non-locally reacting boundaries and nonlinear acoustic boundary condition. We prove the existence and uniqueness of solution. Additionally, using the method of Nakao we demonstrate the decay of the associated energy (Stability).

Nonlinear wave equation in frequency domain: accurate modeling of ultrafast interaction in anisotropic nonlinear media

We interpret the purely spectral forward Maxwell equation with up to 3${^{\rm rd}}$ order induced polarizations for pulse propagation and interactions in quadratic nonlinear crystals. The interpreted equation, also named nonlinear wave equation in frequency domain, includes both quadratic and cubic nonlinearities, delayed Raman effects and anisotropic nonlinearities. The full potential of this wave equation is demonstrated by investigating simulations of solitons generated in the process of ultrafast cascaded second-harmonic generation. We show that a balance in the soliton delay can be achieved due to competition between self-steepening, Raman effects and self-steepening-like effects from cascading originating in the group-velocity mismatch between the pump and second harmonic. We analyze the first-order contributions, and show that this balance can be broken to create fast or slow pulses. Through further simulations we demonstrate few-cycle compressed solitons in extremely short crystals, where spectral phenomena such as blue/red shifting, non-stationary radiation in accordance with the non-local phase matching condition and dispersive-wave generation are observed and marked, which help improving the experimental knowledge of cascading nonlinear soliton pulse compression.