High-order Absorbing Boundary Conditions Research Articles

Compact high order schemes with gradient-direction derivatives for absorbing boundary conditions

We consider several compact high order absorbing boundary conditions (ABCs) for the Helmholtz equation in three dimensions. A technique called “the gradient method” (GM) for ABCs is also introduced and combined with the high order ABCs. GM is based on the principle of using directional derivatives in the direction of the wavefront propagation. The new ABCs are used together with the recently introduced compact sixth order finite difference scheme for variable wave numbers. Experiments on problems with known analytic solutions produced very accurate results, demonstrating the efficacy of the high order schemes, particularly when combined with GM. The new ABCs are then applied to the SEG/EAGE Salt model, showing the advantages of the new schemes.

The double absorbing boundary method for elastodynamics in homogeneous and layered media

Abstract Background Recently the Double Absorbing Boundary (DAB) method was introduced as a new approach for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. Methods The DAB method is based on truncating the unbounded domain to produce a finite computational domain, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and within the layer, and participate in the numerical scheme. In previous studies DAB was developed for acoustic waves which are solutions to the scalar wave equation. Here the approach is extended to time-dependent elastic waves in homogeneous and layered media. The equations are written in second-order form in space and time. Standard Finite Elements (FE) are used for space discretization and the damped Newmark scheme is used for time discretization. Results The performance of the scheme is demonstrated via numerical examples. The DAB was applied to elastodynamics problems in conjunction with the FE method to demonstrate the performance of the method. Conclusions DAB is a viable method for solving wave problems in unbounded domains.

Application of the double absorbing boundary condition in seismic modeling

We apply the newly proposed double absorbing boundary condition (DABC) (Hagstrom et al., 2014) to solve the boundary reflection problem in seismic finite-difference (FD) modeling. In the DABC scheme, the local high-order absorbing boundary condition is used on two parallel artificial boundaries, and thus double absorption is achieved. Using the general 2D acoustic wave propagation equations as an example, we use the DABC in seismic FD modeling, and discuss the derivation and implementation steps in detail. Compared with the perfectly matched layer (PML), the complexity decreases, and the stability and flexibility improve. A homogeneous model and the SEG salt model are selected for numerical experiments. The results show that absorption using the DABC is considerably improved relative to the Clayton-Engquist boundary condition and nearly the same as that in the PML.

Comments on iterative schemes for high order compact discretizations to the exterior Helmholtz equation

We consider various formulations of higher order absorbing boundary conditions for the Helmholtz equation.

Optimized Schwarz Method with Complete Radiation Transmission Conditions for the Helmholtz Equation in Waveguides

In this paper, we present a nonoverlapping optimized Schwarz algorithm with a high-order transmission condition for the Helmholtz equation posed in waveguides. We introduce the new high-order transmission conditions based on the complete radiation boundary conditions (CRBCs) that have been developed for high-order absorbing boundary conditions. We obtain the convergence rate of the algorithm in terms of reflection coefficients of CRBCs, which decrease exponentially with the order of CRBCs. It will be shown that damping parameters involved in the transmission conditions can be selected in an optimal way for enhancing the convergence of the Schwarz algorithm. Also, this algorithm can be employed efficiently for a preconditioner in GMRES implementations based on the substructured form as in [M. J. Gander, F. Magoulès, and F. Nataf, SIAM J. Sci. Comput., 24 (2002), pp. 38--60]. Finally, numerical examples confirming the theory will be presented.

Domain Decomposition Method and High-Order Absorbing Boundary Conditions for the Numerical Simulation of the Time Dependent Schrödinger Equation with Ionization and Recombination by Intense Electric Field

This paper is devoted to the efficient computation of the time dependent Schrödinger equation for quantum particles subject to intense electromagnetic fields including ionization and recombination of electrons with their parent ion. The proposed approach is based on a domain decomposition technique, allowing a fine computation of the wavefunction in the vicinity of the nuclei located in a domain $$\Omega _1$$ and a fast computation in a roughly meshed domain $$\Omega _2$$ far from the nuclei where the electrons are assumed free. The key ingredients in the method are (i) well designed transmission boundary conditions on $$\partial \Omega _1$$ (resp. $$\partial \Omega _2$$ ) in order to estimate the part of the wavefunction “leaving” Domain $$\Omega _1$$ (resp. $$\Omega _2$$ ), (ii) a Schwarz waveform relaxation algorithm to accurately reconstruct the solution. The developed method makes it possible for electrons to travel from one domain to another without loosing accuracy, when the frontier or the overlapping region between two domains is crossed by the wavefunction.

Double Absorbing Boundary Formulations for Acoustics and Elastodynamics

Wave problems in unbounded domains are often treated numerically by truncating the domain to produce a finite computational domain. The double absorbing boundary (DAB) method, which was invented recently as an alternative to methods of high-order absorbing boundary conditions and to the perfectly matched layer, is investigated here for problems in acoustics and elastodynamics. The paper offers two main contributions. The first one pertains to the well-posedness of the DAB scheme for the acoustics problem written in second-order form. The energy method is employed to obtain uniform-in-time estimates of the norm of the solution and the auxiliary functions, thus establishing the well-posedness and asymptotic stability of the DAB formulation. The second part pertains to the extension of the DAB to isotropic elastodynamics, written in first-order conservation form. Numerical experiments for an elastic wave guide demonstrate the performance of the scheme.

The Double Absorbing Boundary method

A new approach is devised for solving wave problems in unbounded domains. It has common features to each of two types of existing techniques: local high-order Absorbing Boundary Conditions (ABC) and Perfectly Matched Layers (PML). However, it is different from both and enjoys relative advantages with respect to both. The new method, called the Double Absorbing Boundary (DAB) method, is based on truncating the unbounded domain to produce a finite computational domain Ω, and on applying a local high-order ABC on two parallel artificial boundaries, which are a small distance apart, and thus form a thin non-reflecting layer. Auxiliary variables are defined on the two boundaries and inside the layer bounded by them, and participate in the numerical scheme. The DAB method is first introduced in general terms, using the 2D scalar time-dependent wave equation as a model. Then it is applied to the 1D Klein–Gordon equation, using finite difference discretization in space and time, and to the 2D wave equation in a wave guide, using finite element discretization in space and dissipative time stepping. The computational aspects of the method are discussed, and numerical experiments demonstrate its performance.

STRESS–VELOCITY COMPLETE RADIATION BOUNDARY CONDITIONS

A new high-order local Absorbing Boundary Condition (ABC) has been recently proposed for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. It is based on the stress–velocity formulation of the elastodynamics problem, and on the general Complete Radiation Boundary Condition (CRBC) approach, originally devised by Hagstrom and Warburton in 2009. The work presented here is a sequel to previous work that concentrated on the stability of the scheme; this is the first known high-order ABC for elastodynamics which is long-time stable. Stability was established both theoretically and numerically. The present paper focuses on the accuracy of the scheme. In particular, two accuracy-related issues are investigated. First, the reflection coefficients associated with the new CRBC for different types of incident and reflected elastic waves are analyzed. Second, various choices of computational parameters for the CRBC, and their effect on the accuracy, are discussed. These choices include the optimal coefficients proposed by Hagstrom and Warburton for the acoustic case, and a simplified formula for these coefficients. A finite difference discretization is employed in space and time. Numerical examples are used to experiment with the scheme and demonstrate the above-mentioned accuracy issues.

A high-order based boundary condition for dynamic analysis of infinite reservoirs

An approach for modeling the radiation damping of an infinite acoustic waveguide is the high-order absorbing boundary condition. This approach is employed in this study as a basis for analysis of an infinite water channel. In order to achieve an accurate and efficient approach for this purpose, the imposed boundary condition should have two features: (a) the reservoir’s bottom absorption which highly affects the pressure distribution inside the reservoir should be included in the formulation of the ABC; (b) the far-field base excitation, which is a significant factor during the vertical excitation should also be considered in the analysis. By separating the overall hydrodynamic pressure into scattered and incident pressure and involving the scattered term in the formulation of the ABC, an accurate and versatile boundary condition is obtained. This boundary condition is then applied in the analysis of several benchmark examples and the obtained results demonstrate the efficiency of the proposed technique.

A general approach for high order absorbing boundary conditions for the Helmholtz equation

When solving a scattering problem in an unbounded space, one needs to implement the Sommerfeld condition as a boundary condition at infinity, to ensure no energy penetrates the system. In practice, solving a scattering problem involves truncating the region and implementing a boundary condition on an artificial outer boundary. Bayliss, Gunzburger and Turkel (BGT) suggested an Absorbing Boundary Condition (ABC) as a sequence of operators aimed at annihilating elements from the solution’s series representation. Their method was practical only up to a second order condition. Later, Hagstrom and Hariharan (HH) suggested a method which used auxiliary functions and enabled implementation of higher order conditions.We compare various absorbing boundary conditions (ABCs) and introduce a new method to construct high order ABCs, generalizing the HH method. We then derive from this general method ABCs based on different series representations of the solution to the Helmholtz equation – in polar, elliptical and spherical coordinates. Some of these ABCs are generalizations of previously constructed ABCs and some are new.These new ABCs produce accurate solutions to the Helmholtz equation, which are much less dependent on the various parameters of the problem, such as the value of k, or the eccentricity of the ellipse. In addition to constructing new ABCs, our general method sheds light on the connection between various ABCs. Computations are presented to verify the high accuracy of these new ABCs.

High-order absorbing boundary conditions for the meshless radial point interpolation method in the frequency domain

The meshless radial point interpolation method (RPIM) in frequency domain for electromagnetic scattering problems is presented. This method promises high accuracy in a simple collocation approach u ...

Long-time stable high-order absorbing boundary conditions for elastodynamics

Two new high-order local absorbing boundary conditions (ABCs) are devised for use on an artificial boundary for time-dependent elastic waves in unbounded domains, in two dimensions. The elastic medium in the exterior domain is assumed to be homogeneous and isotropic. The first ABC is written directly with respect to the displacement vector field, using an operator involving high derivatives. The second ABC is applied to the problem written in a first-order conservation form, using stresses and velocities as variables, and is formulated as a sequence of recursive relations using auxiliary variables. The two ABCs are not equivalent but are closely related. In each case, the order of the ABC determines its accuracy and can be chosen to be arbitrarily high. Both ABCs involve a product of first-order differential operators; all of them are of the Higdon type, except one which is of the Lysmer–Kuhlemeyer type. The stability of both ABCs is analyzed theoretically. The second (stress–velocity) ABC is implemented, employing a finite difference discretization scheme in space and time. Numerical experiments demonstrate the performance of the scheme. To the best of the authors’ knowledge, these are the first existing local high-order ABCs for elastodynamics which are long-time stable.

Numerical comparison of high-order absorbing boundary conditions and perfectly matched layers for a dispersive one-dimensional medium

High-order absorbing boundary conditions (ABC) and perfectly matched layers (PML) are two powerful methods to numerically solve wave problems in unbounded domains. The aim of the proposed study is to analyze and compare the performance of these methods in the one-dimensional problem governed by the dispersive wave equation. The PMLs proposed in literature for time-harmonic dynamics are applied to time-dependent wave problems, and linear, quadratic and cubic polynomial stretching functions are considered. The resulting PMLs exhibit a double absorbing action: (i) they reduce the amplitude of the incident waves and (ii) they delay the wave propagation. Then the ABC proposed by Givoli and Neta and those proposed by Hagstrom, Mar-Or and Givoli are considered. The former are a reformulation of the Higdon high-order non-reflecting boundary conditions, the latter improve the Higdon conditions and extend them to take into account evanescent waves. The accuracy of the PMLs and ABCs, implemented in a finite element code, is first investigated with respect to the frequency of the incident wave, being it progressive or evanescent. Then the response to a wave train characterized by a broad frequency spectrum, resulting from an impulsive force, is studied. A detailed analysis is performed to detect the influence of the parameters of both the ABCs and the PMLs on the absorption of waves. The performances of PMLs and ABCs are compared, and merits and drawbacks of the two methods are pointed out.

Analysis of High-Order Absorbing Boundary Conditions for the Schrödinger Equation

AbstractThe paper is concerned with the numerical solution of Schrödinger equations on an unbounded spatial domain. High-order absorbing boundary conditions for one-dimensional domain are derived, and the stability of the reduced initial boundary value problem in the computational interval is proved by energy estimate. Then a second order finite difference scheme is proposed, and the convergence of the scheme is established as well. Finally, numerical examples are reported to confirm our error estimates of the numerical methods.

A finite element scheme with a high order absorbing boundary condition for elastodynamics

A high-order absorbing boundary condition (ABC) is devised on an artificial boundary for time-dependent elastic waves in unbounded domains. The configuration considered is that of a two-dimensional elastic waveguide. In the exterior domain, the unbounded elastic medium is assumed to be isotropic and homogeneous. The proposed ABC is an extension of the Hagstrom–Warburton ABC which was originally designed for acoustic waves, and is applied directly to the displacement field. The order of the ABC determines its accuracy and can be chosen to be arbitrarily high. The initial boundary value problem including this ABC is written in second-order form, which is convenient for geophysical finite element (FE) analysis. A special variational formulation is constructed which incorporates the ABC. A standard FE discretization is used in space, and a Newmark-type scheme is used for time-stepping. A long-time instability is observed, but simple means are shown to dramatically postpone its onset so as to make it harmless during the simulation time of interest. Numerical experiments demonstrate the performance of the scheme.

High‐order absorbing boundary conditions incorporated in a spectral element formulation

AbstractThe solution of the time‐dependent wave equation in a semi‐infinite wave guide is considered. An artificial boundary ℬ︁ is introduced, which encloses a finite computational domain. On ℬ︁ the Hagstrom–Warburton (H–W) local high‐order absorbing boundary condition (ABC) is imposed. In contrast to previous studies, which involved either finite difference schemes or low‐order finite element schemes, here the way to incorporate the H–W ABC into a spectral element formulation is shown. To this end, the Seriani–Priolo spectral element is used in space and a 4th‐order Runge–Kutta scheme is employed in time. This leads to exponential convergence in both the polynomial order of the spectral element, and in the order of the ABC, and thus avoids the convergence rate inconsistency which is otherwise present. The combined ABC‐spectral formulation enables one to obtain extremely accurate solutions of wave problems in unbounded domains. This is demonstrated via a number of numerical examples. Copyright © 2008 John Wiley & Sons, Ltd.

High-order Absorbing Boundary Conditions for anisotropic and convective wave equations

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. They are extensions of the construction of Hagstrom, Givoli and Warburton for the isotropic stationary case. These ABCs are local, and involve only low-order derivatives owing to the use of auxiliary variables on the artificial boundary. The accuracy and well-posedness of these ABCs is analyzed. Special attention is given to the issue of mismatch between the directions of phase and group velocities, which is a potential source of concern. Numerical examples for the anisotropic case are presented, using a finite element scheme.

Implementation of higher-order absorbing boundary conditions for the Einstein equations

We present an implementation of absorbing boundary conditions for the Einstein equations based on the recent work of Buchman and Sarbach. In this paper, we assume that spacetime may be linearized about Minkowski space close to the outer boundary, which is taken to be a coordinate sphere. We reformulate the boundary conditions as conditions on the gauge-invariant Regge–Wheeler–Zerilli scalars. Higher-order radial derivatives are eliminated by rewriting the boundary conditions as a system of ODEs for a set of auxiliary variables intrinsic to the boundary. From these we construct boundary data for a set of well-posed constraint-preserving boundary conditions for the Einstein equations in a first-order generalized harmonic formulation. This construction has direct applications to outer boundary conditions in simulations of isolated systems (e.g., binary black holes) as well as to the problem of Cauchy-perturbative matching. As a test problem for our numerical implementation, we consider linearized multipolar gravitational waves in TT gauge, with angular momentum numbers ℓ = 2 (Teukolsky waves), 3 and 4. We demonstrate that the perfectly absorbing boundary condition of order L = ℓ yields no spurious reflections to linear order in perturbation theory. This is in contrast to the lower-order absorbing boundary conditions with L < ℓ, which include the widely used freezing-Ψ0 boundary condition that imposes the vanishing of the Newman–Penrose scalar Ψ0.

Efficient High-Order Absorbing Boundary Condition for the ADI-FDTD Method

In this letter, an efficient high-order absorbing boundary condition for the alternating-direction-implicit-finite-difference time-domain (ADI-FDTD) method is developed. The proposed high-order absorbing boundary condition (ABC) has very high numerical efficiency because it can keep the tri-diagonal matrix form during the ADI iteration. The absorbing performance of the proposed ABC is analyzed theoretically. To verify the performance of the proposed high-order ABC, a rectangular waveguide problem which has very strong dispersion characteristic is analyzed with the ADI-FDTD method. Numerical results show that the proposed high-order ABC has excellent absorbing performance.