# Well-posedness of one-way wave equations and absorbing boundary conditions

by Lloyd N. Trefethen

Publisher: Institute for Computer Applications in Science and Engineering, NASA Langley Research Center, Publisher: For sale by the National Technical Information Service in Hampton, Va, [Springfield, Va

Written in English

## Subjects:

• Wave equation.

## Edition Notes

The Physical Object ID Numbers Other titles Well posedness of one way wave equations and absorbing boundary conditions. Statement Lloyd N. Trefethen, Laurence Halpern. Series ICASE report -- no. 85-30., NASA contractor report -- 172619., NASA contractor report -- NASA CR-172619. Contributions Halpern, Laurence., Institute for Computer Applications in Science and Engineering. Format Microform Pagination 1 v. Open Library OL18029815M

which imply the one-way equations, i n x,t t = Tˆ + Vˆ x,t. 16 They approximate Eq. 10 at two boundaries and can act as absorbing boundary conditions we need. Concretely, we ob-tain nonlinear absorbing boundary conditions: n =2: i t = − i 2 m k 0 hand x + k2 + V x + f 2, 17 n =3: i 2i x 6k 0 t = − 2 m 3ik2 x k3 + V x + f 2 2i x 6k 0. equation and to derive a nite ﬀ approximation to the heat equation. Similarly, the technique is applied to the wave equation and Laplace’s Equation. The technique is illustrated using EXCEL spreadsheets. Key Concepts: Finite ﬀ Approximations to derivatives, The Finite ﬀ Method, The Heat Equation, The Wave Equation, Laplace’s Equation. Solution to Problems for the 1-D Wave Equation Linear Partial Diﬀerential Equations (i.e. we don’t worry about boundary conditions), what initial Hence we know (7) satisﬁes the wave equation, by the way we found D’Alembert’s File Size: KB.   In the present paper, an absorbing boundary condition is presented for Lattice Boltzmann equations (LBE) which is formulated based on the PML concept. A brief introduction to the LBM is presented in Sec. The theoretical background on the PML formulation for general hyperbolic equations is presented in Sec. by:

Absorbing Boundary Conditions and Numerical Methods for the Linearized Water Wave Equation in 1 and 2 Dimensions by David K. Prigge A dissertation submitted in partial ful llment of the requirements for the degree of Doctor of Philosophy (Applied and Interdisciplinary Mathematics) in The University of Michigan Doctoral Committee. The aim of the current work is to provide an easy-to-apply algorithm to determine the correct type and number of boundary conditions for first order hyperbolic systems of equations by providing a necessary condition between the characteristic variables and the primitive variables at the boundary Cited by: 6. In [29, 33], Szeftel designed absorbing boundary conditions for one-dimensional nonlinear wave equation by the potential and the paralinear strategies. Soﬀer and Stucchio pre-sented a phase space ﬁlter method to obtain absorbing boundary conditions. The PML [13, 39] was also applied to handling the nonlinear Schro¨dinger equations. One-way wave equations (OWWEs), derived from rational approximations, C(s) to 1/√1 - s 2, are ing boundary conditions obtained from these OWWEs are easily implemented, producing systems of differential equations at the boundary which are different from those produced by rational approximations, r(s) to √1 - s gh these systems are different, a particular choice of Author: R. A. Renaut, J. S. Parent.

The boundary condition () is the desired absorbing tmindary condit,iori for this problem, which is exact. We note that this condition could have been applieci wactly at s = I,. anti this. as WP will see. cloes not always hold in higtier'ciiriiensions. There is another tccliriic*id clif1icrilty in this type of Size: 1MB. Boundary Conditions for the Wave Equation We now consider a nite vibrating string, modeled using the PDE u tt = c2u xx; 0 0 and initial conditions u(x;0) = f(x); u t(x;0) = g(x); 0 boundary conditions of one of the following three forms: 1. Controlled end points: When the ends of the string are speci ed, we. Ordinary differential equations: theory and practice: an elementary, integrated, applied treatment bas The modulation of a weekly nonlinear wave pocket in an inhomogenous medium / by R. Grinshaw; Well-posedness of one-way wave equations and absorbing boundary conditions [microform] / Lloyd N. . 25 Problems: Separation of Variables - Heat Equation 26 Problems: Eigenvalues of the Laplacian - Laplace 27 Problems: Eigenvalues of the Laplacian - Poisson 28 Problems: Eigenvalues of the Laplacian - Wave 29 Problems: Eigenvalues of the Laplacian - Heat Heat Equation with Periodic Boundary Conditions in 2D.

## Well-posedness of one-way wave equations and absorbing boundary conditions by Lloyd N. Trefethen Download PDF EPUB FB2

Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions By Lloyd N. Trefethen* and Laurence l (aipern Abstract. A one-way wave equation is a partial differential equation which, in some approxi-mate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one.

Well-Posedness of One-Way Wave Equations and Absorbing Boundary Conditions By Lloyd N. Trefethen* and Laurence Halpern Abstract.

A one-way wave equation is a partial differential equation whlch, in some approxi- mate sense, behaves like the wave equation in one direction but permits no propagation in the opposite one. Well-posedness of one-way wave equations and absorbing boundary conditions Author: Lloyd N Trefethen ; Laurence Halpern ; Institute for Computer Applications in Science and Engineering.

Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the wave equation.

We find that if r(s) interpolates the square root of (1-s sup 2) at sufficiently many points in (-1,1), then well-posedness is assured. The one-way wave equations occurring in geophysics, underwater acoustics, and numerical studies involving absorbing boundary conditions are Well-posedness of one-way wave equations and absorbing boundary conditions book analytically.

The conditions under which such equations are well posed are obtained by examining the rational functions used to reduce : Lloyd N. Trefethen and Laurence Halpern. Halpern L., Rahmouni A. () One-way operators, absorbing boundary conditions and domain decomposition for wave propagation.

In: Bourlioux A., Gander M.J., Sabidussi G. (eds) Modern Methods in Scientific Computing and Applications. NATO Science Series (Series II: Mathematics, Physics and Chemistry), vol Springer, DordrechtCited by: 2. in issue.

Absorbing Boundary Conditions for the Elastic Wave Equations James Sochacki Department of Mathematics University of California, Davis Davis, California ABSTRACT The two dimensional elastic wave equations are used to model wave propagation in mediums with large or unbounded domains. In order to numerically simulate those problems the equations have to be put Cited by: In this work, new absorbing boundary conditions (ABCs) for a wave equation with a temperature-dependent speed of sound are proposed.

Based on the theory of pseudo-differential calculus, first- and second-order ABCs for the one- and two-dimensional wave equations are derived. Both boundary conditions are local in space and by: 1. Keywords: absorbing boundary conditions, one-way wave equations, well-posedness, stability, dis-crete approximations.

Absorbing Boundary Conditions are boundary procedures that are applied at the arti cial numerical boundaries of a computational domain to miminize or eliminate the spurious relections at these boundaries which occur in theFile Size: 42KB.

ity of solutions is established in Section and thereby completing the proof of well-posedness of Cauchy problem. In Sectioninitial boundary value problems are considered for one dimensional wave equation. Existence of solutions Inthissection,wederiveanexpressionforsolutiontothehomogeneousCauchyproblemFile Size: KB.

In absorbing boundary condItion apphcatIOns, the domam is x, t > 0, y E R, and the one-way wave equatIOn IS applIed as a boundary condItion along X = 0 for (11) Well-posedness IS now the eXIstence of a UnIque solutIOn whose norm at t = to and along X = 0 can be estImated m terms of the. Well-posedness of one-way wave equations and absorbing boundary conditions Lloyd N.

Trefethen and Laurence Halpern. Math. Comp. 47 (), Abstract, references and article information Full-text PDF Free Access Request permission to use this material MathSciNet review: Well-posedness of the Westervelt equation with higher order absorbing boundary conditions - ScienceDirect.

: Barbara Kaltenbacher, Igor Shevchenko. A technique that has proven successful is the application of absorbing boundary conditions which have been derived from approximations to a one-way wave equation (OWWE) at the bound- ary [2, 5, 7, 13, 9].

In this paper we reconsider this approach and show that the methods presented. A one-way wave equation, also known as a paraxial or parabolic wave equation, is a differential equation that permits wave propagation in certain directions only.

Such equations are used regularly in underwater acoustics, in geophysics, and as energy-absorbing numerical boundary conditions. The design of a one-way wave equation is connected. High-order Absorbing Boundary Conditions (ABCs), applied on a rectangular artificial computational boundary that truncates an unbounded domain, are constructed for a general two-dimensional linear scalar time-dependent wave equation which represents acoustic wave propagation in anisotropic and subsonically convective media.

Previously published absorbing boundary conditions will be shown to reduce to special cases of this absorbing boundary condition. The well-posedness of the initial boundary value problem of the absorbing boundary condition, coupled to the interior Schrödinger equation, will also be by: A one‐way wave equation, also known as a paraxial or parabolic wave equation, is a differential equation that permits wave propagation in certain directions only.

Such equations are used regularly in underwater acoustics, in geophysics, and as energy‐absorbing numerical boundary by: The Higdon sequence of Absorbing Boundary Conditions (ABCs) for the linear wave equation is considered. Building on a previous work of Ha-Duong and Joly, which related to other forms of boundary conditions, the Higdon ABCs are proved to be energy-stable (on the continuous level) up to any : BaffetDaniel, GivoliDan.

Traditional boundary conditions describe the interaction of the isolated system we are modeling with the rest of the physical world. [For example, there may be perfect insulation at the ends of a conducting bar if we are solving the heat equation; see Churchill () for a discussion.] For meteorological limited-area modeling, however, there is no physical by: Those rational functions r for which the corresponding one-way wave equation is well-posed are characterized both as a partial differential equation and as an absorbing boundary condition for the Author: Jeremie Szeftel.

Trefethen L. N., Halpern N. Well-posedness of one-way wave equations and absorbing boundary conditions. Math. Comp., v,pp. – Google ScholarCited by: 3. () Well-posedness of the Westervelt equation with higher order absorbing boundary conditions.

Journal of Mathematical Analysis and Applications() Mathematical analysis of Ziolkowski’s PML model with application for wave propagation in by: The mathematics of PDEs and the wave equation There are one way wave equations, and the general solution to the two way equation could be done by forming linear combinations of such solutions.

The solutions of the one wave Choosing which solution is a question of initial conditions and boundary values. In fact,Cited by: 2. BOUNDARY CONDITIONS FOR THE WAVE EQUATION 67 These remarks generally do not apply when one-way wave equations of order two or more are used as absorbing boundary conditions but are not factored as described above.

(See, e.g., Section 8 of .) The analysis in the present paper is performed using geometrical constructions in. Key words. Helmholtz equation, waveguide, nonlocal boundary conditions, a priori estimates.

AMS subject classi cations. 35J05, 35J20, 65N30, 76Q05 1. Introduction. In this paper we develop and analyze a model for wave propagation based on the Helmholtz equation in the context of a realistic environment widely used in applicationsFile Size: KB.

domain. One way out of this problem is to truncate the large domain and to equip () or() with so-calledabsorbing boundary conditions. Recently, Kaltenbacher & Shevchenko [14, 24] derived and proposed absorbing boundary conditions of order zero and order one for the Westervelt equation () in one and two space : Gieri Simonett, Mathias Wilke.

Radiation boundary conditions for the numerical simulation of waves - Volume 8 - Thomas Hagstrom Petropoulos, P. (), ‘ Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell's equations in rectangular, (), ‘ Well-posedness of one-way wave equations and absorbing boundary Cited by: We reconsider the stability theory of boundary conditions for the wave equation from the point of view of energy techniques.

We study, for the case of the homogeneous half-space, a large class of boundary conditions including the so-called absorbing conditions. We show that the results of strong stability in the sense of Kreiss, studied from the point of view of the modal analysis by Trefethen.

This book presents the key ideas along with many figures, examples, and short, elegant MATLAB programs for readers to adapt to their own needs.

Well-posedness of one-way wave equations and absorbing boundary conditions by Lloyd N Trefethen Defines the fundamental boundary conditions governing the effects of these forces.

Includes. The Equation of Motion and Boundary Conditions The wave equation is a second-order linear partial differential equation u tt = c2∆u+f (1) with u tt = ∂2u ∂t 2, ∆ = ∇∇ = ∂ 2 ∂x + ∂ ∂y + ∂ ∂z2, (2) whese u is the pressure ﬁeld (as described above) and c is the speed of sound, which we assume to be constant in the File Size: KB.The implementation of the "absorbing boundary conditions" (ABCs) for the far-field boundary based on the wave equations was developed by Engquist-Halpern .

In addition, the large-time ABCs for.It is these reflections that are to be minimized. A technique that has proven successful, is the application of absorbing boundary conditions which have been derived from approximations to a one-way wave equation (OWWE) at the boundary ,,, ,.Cited by: