Wave Equation In Unbounded Domains Research Articles

Existence of time-dependent attractor of wave equation in locally uniform space

In this article, we study non-autonomous dynamical behavior of weakly damped wave equation in unbounded domain. First of all, we introduce the time-dependent locally uniform space. After that, the pullback asymptotical compactness is proved by applying the contractive function method. Eventually, we obtain the existence of ( H lu 1 ( R 3 ) × L lu 2 ( R 3 ) , H ρ 1 ( R 3 ) × L ρ 1 ( R 3 ) ) -time-dependent attractor of wave equation.

Unconditionally energy-stable mapped Gegenbauer spectral Galerkin method for diffusive–viscous wave equation in unbounded domains

The diffusive–viscous wave equation plays a significant role in investigating the attenuation of seismic wave propagating in fluid-saturated medium. This paper focuses on the design of unconditionally energy-stable spectral method for the diffusive–viscous wave equation defined in unbounded domains Rd(d=2,3). We develop such a spectral method by using mapped Gegenbauer functions for the spatial approximation and Crank–Nicolson scheme for the temporal discretization. Then we show that the fully-discrete method satisfies the discrete energy-dissipation law without any restriction on the time step size. To achieve an efficient implementation, the matrix diagonalization procedure is employed to solve the linear systems for d=2,3. Finally, we carry out numerical examples in 3D case to demonstrate the good behavior of our method.

Two efficient spectral methods for the nonlinear fractional wave equation in unbounded domain

This paper is concerned with two fast and efficient numerical methods to solve the multidimensional nonlinear fractional wave equation in unbounded domain. For the spatial discretization, a spectral-Galerkin method is adopted by using the Fourier-Like Mapped Chebyshev function approximation, which makes fractional Laplacian diagonalized. Then, for the temporal discretization, the second order accurate time-splitting method is proposed and it not only avoids the matrix exponential computation in view of the diagonalized structure of semi-discrete systems, but also results in an explicit and time symmetric scheme; Then, an exponential scalar auxiliary variable (E-SAV) method is developed for solving fractional wave equation, which preserves the original energy conservation and makes nonlinear term to be solved explicitly to obtain a linear system at each time step. In addition, the two schemes both can be implemented by the fast Fourier transform (FFT) related to Chebyshev polynomials. Numerical experiments are provided to test the numerical accuracy and the efficiency in long time simulations.

Uniqueness to Some Inverse Source Problems for the Wave Equation in Unbounded Domains

This paper is concerned with inverse acoustic source problems in an unbounded domain with dynamical boundary surface data of Dirichlet kind. The measurement data are taken at a surface far away from the source support. We prove uniqueness in recovering source terms of the form f(x)g(t) and f(x1, x2, t)h(x3), where g(t) and h(x3) are given and x = (x1, x2, x3) is the spatial variable in three dimensions. Without these a priori information, we prove that the boundary data of a family of solutions can be used to recover general source terms depending on both time and spatial variables. For moving point sources radiating periodic signals, the data recorded at four receivers are prove sufficient to uniquely recover the orbit function. Simultaneous determination of embedded obstacles and source terms was verified in an inhomogeneous background medium using the observation data of infinite time period. Our approach depends heavily on the Laplace transform.

Exponential decay for the damped wave equation in unbounded domains

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilization and control.

On a Reformulated Convolution Quadrature Based Boundary Element Method

Boundary Element formulations in time domain suffer from two prob- lems. First, for hyperbolic problems not too much fundamental solutions are avail- able and, second, the time stepping procedure is expensive in storage and has stabil- ity problems for badly chosen time step sizes. The first problem can be overcome by using the Convolution Quadrature Method (CQM) for time discretisation. This as well improves the stability. However, still the storage requirements are large. A recently published reformulation of the CQM by Banjai and Sauter (Rapid solu- tion of the wave equation in unbounded domains, SIAM J. Numer. Anal., 47, 227- 249) reduces the time stepping procedure to the solution of decoupled problems in Laplace domain. This new version of the CQM is applied here to elastodynam- ics. The storage is reduced to nearly the amount necessary for one calculation in Laplace domain. The properties of the original method concerning stability in time are preserved. Further, the only parameter to be adjusted is still the time step size. The drawback is that the time history of the given boundary data has to be known in advance. These conclusions are validated by the examples of an elastodynamic column and a poroelastodynamic half space.

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

A Galerkin method for retarded boundary integral equations with smooth and compactly supported temporal basis functions

We consider retarded boundary integral formulations of the three-dimensional wave equation in unbounded domains. Our goal is to apply a Galerkin method in space and time in order to solve these problems numerically. In this approach the computation of the system matrix entries is the major bottleneck. We will propose new types of finite-dimensional spaces for the time discretization. They allow variable time-stepping, variable order of approximation and simplify the quadrature problem arising in the generation of the system matrix substantially. The reason is that the basis functions of these spaces are globally smooth and compactly supported. In order to perform numerical tests concerning our new basis functions we consider the special case that the boundary of the scattering problem is the unit sphere. In this case explicit solutions of the problem are available which will serve as reference solutions for the numerical experiments.

Energy decay of solutions to the Cauchy problem for a nondissipative wave equation

In this paper, we consider energy decay for nondissipative wave equation in unbounded domain (the usual energy is not decreasing). We use an approach introduced by Guesmia, which leads to decay estimates (known in the dissipative case) when the integral inequalities method by Komornik [Exact Controllability and Stabilization. The Multiplier Method. Masson–Wiley, Paris, 1994] cannot be applied due to the lack of dissipativity. First, we study the stability of a wave equation with a weak nonlinear dissipative term based on the equation: \documentclass[12pt]{minimal}\begin{document}$u^{\prime \prime }-\Delta _{x}u+\lambda ^{2}(x) u+\sigma (t) g(u^{\prime })+ \theta (t)h(\nabla _{x} u)=0$\end{document}u′′−Δxu+λ2(x)u+σ(t)g(u′)+θ(t)h(∇xu)=0 in \documentclass[12pt]{minimal}\begin{document}${\mathchoice{\bf \hbox{$\displaystyle \rm I \hspace{-1.5pt}\rm R$}}{\bf \hbox{$\textstyle {\rm I} \hspace{-1.5pt}\rm R$}}{\bf \hbox{$\scriptstyle \rm I\hspace{-1.0pt}\rm R$}}{\bf \hbox{$\scriptscriptstyle \rm I \hspace{-1.0pt}\rm R$}}}^{n}$\end{document}IRn. We consider the general case with a function h satisfying a smallness condition, and we obtain decay of solutions under weak growth assumptions on the feedback function and without any control of the sign of the derivative of the energy related to the above equation. In the second case, we consider the case h(∇u) = −∇Φ∇u. We prove some precise decay estimates of equivalent energy.

Rapid Solution of the Wave Equation in Unbounded Domains

In this paper we propose and analyze a new, fast method for the numerical solution of time domain boundary integral formulations of the wave equation. We employ Lubich's convolution quadrature method for the time discretization and a Galerkin boundary element method for the spatial discretization. The coefficient matrix of the arising system of linear equations is a triangular block Toeplitz matrix. Possible choices for solving the linear system arising from the above discretization include the use of fast Fourier transform (FFT) techniques and the use of data-sparse approximations. By using FFT techniques, the computational complexity can be reduced substantially while the storage cost remains unchanged and is, typically, high. Using data-sparse approximations, the gain is reversed; i.e., the computational cost is (approximately) unchanged while the storage cost is substantially reduced. The method proposed in this paper combines the advantages of these two approaches. First, the discrete convolution (related to the block Toeplitz system) is transformed into the (discrete) Fourier image, thereby arriving at a decoupled system of discretized Helmholtz equations with complex wave numbers. A fast data-sparse (e.g., fast multipole or panel-clustering) method can then be applied to the transformed system. Additionally, significant savings can be achieved if the boundary data are smooth and time-limited. In this case the right-hand sides of many of the Helmholtz problems are almost zero, and hence can be disregarded. Finally, the proposed method is inherently parallel. We analyze the stability and convergence of these methods, thereby deriving the choice of parameters that preserves the convergence rates of the unperturbed convolution quadrature. We also present numerical results which illustrate the predicted convergence behavior.

Instability results for the damped wave equation in unbounded domains

We extend some previous results for the damped wave equation in bounded domains in R d to the unbounded case. In particular, we show that if the damping term is of the form αa with bounded a taking on negative values on a set of positive measure, then there will always exist unbounded solutions for sufficiently large positive α. In order to prove these results, we generalize some existing results on the asymptotic behaviour of eigencurves of one-parameter families of Schrödinger operators to the unbounded case, which we believe to be of interest in their own right.

Stabilization of solutions of the first mixed problem for the wave equation in domains with non-compact boundaries

In this paper we study the rate of decay, for large values of time, of the local energy of solutions of the first mixed problem for the wave equation in unbounded domains , , with smooth non-compact boundaries. Under the assumption that the boundary surface satisfies a condition generalizing the condition of star-shapedness with respect to the origin we establish a power estimate of the rate of decay of the local energy as . The proof is based on uniform estimates in the half-plane of solutions of the corresponding spectral problem — the first boundary-value problem for the Helmholtz equation — obtained in the paper.

Fast Quadrature Techniques for Retarded Potentials Based on TT/QTT Tensor Approximation

AbstractWe consider the Galerkin approach for the numerical solution of retarded boundary integral formulations of the three dimensional wave equation in unbounded domains. Recently smooth and compactly supported basis functions in time were introduced which allow the use of standard quadrature rules in order to compute the entries of the boundary element matrix. In this paper, we use TT and QTT tensor approximations to increase the efficiency of these quadrature rules. Various numerical experiments show the substantial reduction of the computational cost that is needed to obtain accurate approximations for the arising integrals.

A simple impedance-infinite element for the finite element solution of the three-dimensional wave equation in unbounded domains

This paper is concerned with the development of an efficient and accurate impedance-infinite element that can be used either in the frequency- or directly in the time-domain for the modeling and solution of problems described by the scalar three-dimensional wave equation in infinite or semi-infinite domains. The infinite domain is truncated and the effect of the truncated infinite region is simulated by the introduction of an absorbing boundary condition prescribed on the truncation boundary. A systematic procedure for the construction of a family of such conditions of increasing accuracy and complexity is developed with explicit formulas given for approximations up to second order. A central feature of this high-order approximation is that it can be expressed, within the context of a finite element formulation, as a set of local infinite elements located at the boundary of the computational domain, with each element defined by a pair of symmetric, time-invariant, stiffness and damping matrices. This makes it possible to incorporate readily the new local boundary element into finite element software developed for purely interior regions, for applications involving steady-state harmonic or transient excitations. Whereas the theory has been developed formally for arbitrary smooth boundary surfaces, here details are provided for ellipsoidal and spherical boundaries. Thus far, only the latter has been implemented and tested in problems involving cavities and rigid scatterers of spherical and cubic shape. Numerical experiments in both the frequency- and time-domain attest to the efficacy and accuracy of the proposed new element.

Stabilization of the solutions of the wave equation in unbounded domains

The behaviour of the solution of the first mixed problem for the wave equation in an unbounded domain with smooth boundary is studied for large values of time. Estimates of the solution of the first boundary-value problem for the Helmholtz equation with spectral parameter in the closed upper half-plane are obtained, and uniqueness of its solution for a parameter on the real line is proved.

Galerkin/least-squares finite element methods for the reduced wave equation with non-reflecting boundary conditions in unbounded domains

Finite element methods are constructed for the reduced wave equation in unbounded domains. Exterior boundary conditions for a computational problem are derived from an exact relation between the solution and its derivatives on an artificial boundary by the DtN method, precluding singular behavior in finite element models. Galerkin and Galerkin/least-squares finite element methods are presented. Model problems of radiation with inhomogeneous Neumann boundary conditions in plane and spherical configurations are employed to design and evaluate the numerical methods in the entire range of propagation and decay. The Galerkin/least-squares method with DtN boundary conditions is designed to exhibit superior behavior for problems of acoustics, providing accurate solutions with relatively low mesh resolution and allowing numerical damping of unresolved waves. General convergence results guarantee the good performance of Galerkin/least-squares methods on all configurations of practical interest. Numerical tests validate these conclusions.

Finite element methods for the helmholtz equation in an exterior domain: Model problems

Finite element methods are presented for the reduced wave equation in unbounded domains. Model problems of radiation with inhomogeneous Neumann boundary conditions, including the effects of a moving acoustic medium, are examined for the entire range of propagation and decay. Exterior boundary conditions for the computational problem over a finite domain are derived from an exact relation between the solution and its derivatives on that boundary. Galerkin, Galerkin/least-squares and Galerkin/gradient least-squares finite element methods are evaluated by comparing errors pointwise and in integral norms. The Galerkin/least-squares method is shown to exhibit superior behavior for this class of problems.

Design and Analysis of Finite Element Methods for the Helmholtz Equation in Exterior Domains

Finite element methods for the reduced wave equation in unbounded domains are presented. A computational problem over a finite domain is formulated by imposing an exact impedance relation at an artificial exterior boundary. Method design is based on a detailed examination of discrete errors in simplified settings, leading to a thorough analytical understanding of method performance. For this purpose, model problems of radiation with inhomogeneous Neumann boundary conditions, including the effects of a moving acoustic medium, are considered for the entire range of propagation and decay. A Galerkin/least-squares method is shown to exhibit superior behavior for this class of problems.

On a nonolinear wave equation in unbounded domains

We study existence and uniqeness of the nonlinear wave equation urn:x-wiley:01611712:media:ijmm473020:ijmm473020-math-0001 in unbounded domains. The above model describes nonlinear wave phenomenon in non‐homogeneous media. Our techniques ivolve fixed point arguments combined with the energy method.

Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains

Some remarks on the local energy decay of solutions of the initial-boundary value problem for the wave equation in unbounded domains