Performance and stability of the double absorbing boundary method for acoustic-wave propagation

Geophysics ◽  
2019 ◽  
Vol 84 (2) ◽  
pp. T59-T72 ◽  
Author(s):  
Toby Potter ◽  
Jeffrey Shragge ◽  
David Lumley
Keyword(s):  
Boundary Layer ◽  
Wave Equation ◽  
Acoustic Wave ◽  
Acoustic Waves ◽  
Spectral Response ◽  
Domain Boundary ◽  
Grid Point ◽  
High Order ◽  
Computational Domain ◽  
Absorbing Boundary

The double absorbing boundary (DAB) is a novel extension to the family of high-order absorbing boundary condition operators. It uses auxiliary variables in a boundary layer to set up cancellation waves that suppress wavefield energy at the computational-domain boundary. In contrast to the perfectly matched layer (PML), the DAB makes no assumptions about the incoming wavefield and can be implemented with a boundary layer as thin as three computational grid-point cells. Our implementation incorporates the DAB into the boundary cell layer of high-order finite-difference (FD) techniques, thus avoiding the need to specify a padding region within the computational domain. We tested the DAB by propagating acoustic waves through homogeneous and heterogeneous 3D earth models. Measurements of the spectral response of energy reflected from the DAB indicate that it reflects approximately 10–15 dB less energy for heterogeneous models than a convolutional PML of the same computational memory complexity. The same measurements also indicate that a DAB boundary layer implemented with second-order FD operators couples well with higher-order FD operators in the computational domain. Long-term stability tests find that the DAB and CPML methods are stable for the acoustic-wave equation. The DAB has promise as a robust and memory-efficient absorbing boundary for 3D seismic imaging and inversion applications as well as other wave-equation applications in applied physics.

ABSORBING LAYER VIA WAVE-EQUATION SPLITTING

2001 ◽  
Vol 09 (04) ◽  
pp. 1347-1354
Author(s):  
JOSÉ M. CARCIONE ◽  
FABIO CAVALLINI
Keyword(s):  
Wave Equation ◽  
Acoustic Waves ◽  
Computational Cost ◽  
Computational Domain ◽  
Absorbing Boundary ◽  
Narrow Strip ◽  
Splitting Technique ◽  
Localized Source ◽  
Absorbing Layer ◽  
Accurate Scheme

Modeling acoustic waves generated by a localized source is always vexed by the nagging problem of spurious reflections and wraparound arising when the wavefront reaches the boundary of the numerical mesh. This difficulty may be circumvented by using a very large computational domain, which is very inefficient, or can be tackled by using some kind of absorbing boundary technique, which has not yet found a universally satisfactory solution. In this work, the wave equation is modified by introducing a term that is nonzero only in a narrow strip near the boundary. Then, a splitting technique permits to compute part of the solution analytically (hence, at no computational cost), while an application of Weyl's formula for the exponential of a matrix leads to a second-order accurate scheme that completes the algorithm. An application to SH seismic wave modeling shows that the performance of the present method is competitive with standard ones. Moreover, there is evidence for a potential application to the modeling of wave propagation in porous media, where stiff differential equations arise.

scholarly journals Modelling damped acoustic waves by a dissipation-preserving conformal symplectic method

2017 ◽  
Vol 473 (2199) ◽  
pp. 20160798 ◽  
Author(s):  
Wenjun Cai ◽  
Huai Zhang ◽  
Yushun Wang
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Acoustic Waves ◽  
Conservation Law ◽  
Heterogeneous Media ◽  
Numerical Dispersion ◽  
Discrete Version ◽  
Symplectic Method ◽  
Ode System ◽  
Strang Splitting

We propose a novel stable and efficient dissipation-preserving method for acoustic wave propagations in attenuating media with both correct phase and amplitude. Through introducing the conformal multi-symplectic structure, the intrinsic dissipation law and the conformal symplectic conservation law are revealed for the damped acoustic wave equation. The proposed algorithm is exactly designed to preserve a discrete version of the conformal symplectic conservation law. More specifically, two subsystems in conjunction with the original damped wave equation are derived. One is actually the conservative Hamiltonian wave equation and the other is a dissipative linear ordinary differential equation (ODE) system. Standard symplectic method is devoted to the conservative system, whereas the analytical solution is obtained for the ODE system. An explicit conformal symplectic scheme is constructed by concatenating these two parts of solutions by the Strang splitting technique. Stability analysis and convergence tests are given thereafter. A benchmark model in homogeneous media is presented to demonstrate the effectiveness and advantage of our method in suppressing numerical dispersion and preserving the energy dissipation. Further numerical tests show that our proposed method can efficiently capture the dissipation in heterogeneous media.

scholarly journals Micro-Differential Boundary Conditions Modelling the Absorption of Acoustic Waves by 2D Arbitrarily-Shaped Convex Surfaces

2012 ◽  
Vol 11 (2) ◽  
pp. 674-690 ◽  
Author(s):  
Hélène Barucq ◽  
Julien Diaz ◽  
Véronique Duprat
Keyword(s):  
Wave Equation ◽  
Acoustic Waves ◽  
Principal Symbol ◽  
Absorbing Boundary Condition ◽  
Acoustic Wave Equation ◽  
Absorbing Boundary ◽  
Computational Burden ◽  
First Order ◽  
Robin Condition ◽  
Elliptic Regions

AbstractWe propose a new Absorbing Boundary Condition (ABC) for the acoustic wave equation which is derived from a micro-local diagonalization process formerly defined by M.E. Taylor and which does not depend on the geometry of the surface bearing the ABC. By considering the principal symbol of the wave equation both in the hyperbolic and the elliptic regions, we show that a second-order ABC can be constructed as the combination of an existing first-order ABC and a Fourier-Robin condition. We compare the new ABC with other ABCs and we show that it performs well in simple configurations and that it improves the accuracy of the numerical solution without increasing the computational burden.

Stability of wide‐angle absorbing boundary conditions for the wave equation

Geophysics ◽  
1989 ◽  
Vol 54 (9) ◽  
pp. 1153-1163 ◽  
Author(s):  
R. A. Renaut ◽  
J. Petersen
Keyword(s):  
Wave Equation ◽  
Boundary Conditions ◽  
Least Squares ◽  
Absorbing Boundary Conditions ◽  
Computational Domain ◽  
Wide Angle ◽  
Courant Number ◽  
Absorbing Boundary ◽  
Wide Range ◽  
Artificial Boundaries

Numerical solution of the two‐dimensional wave equation requires mapping from a physical domain without boundaries to a computational domain with artificial boundaries. For realistic solutions, the artificial boundaries should cause waves to pass directly through and thus mimic total absorption of energy. An artificial boundary which propagates waves in one direction only is derived from approximations to the one‐way wave equation and is commonly called an absorbing boundary. Here we investigate order 2 absorbing boundary conditions which include the standard paraxial approximation. Absorption properties are compared analytically and numerically. Our numerical results confirm that the [Formula: see text] or Chebychev‐Padé approximations are best for wide‐angle absorption and that the Chebychev or least‐squares approximations are best for uniform absorption over a wide range of incident angles. Our results also demonstrate, however, that the boundary conditions are stable for varying ranges of Courant number (ratio of time step to grid size). We prove that there is a stability barrier on the Courant number specified by the coefficients of the boundary conditions. Thus, proving stability of the interior scheme is not sufficient. Furthermore, waves may radiate spontaneously from the boundary, causing instability, even if the stability bound on the Courant number is satisfied. Consequently, the Chebychev and least‐squares conditions may be preferred for wide‐angle absorption also.

scholarly journals Flow in the exit of open pipes during acoustic resonance

1980 ◽  
Vol 99 (2) ◽  
pp. 293-319 ◽  
Author(s):  
J. H. M. Disselhorst ◽  
L. Van Wijngaarden
Keyword(s):  
Boundary Layer ◽  
Mathematical Model ◽  
Boundary Condition ◽  
Acoustic Wave ◽  
Acoustic Waves ◽  
Acoustic Radiation ◽  
Acoustic Resonance ◽  
The Other ◽  
Open Tube ◽  
Theoretical Predictions

The flow near the mouth of an open tube is examined, experimentally and theoretically, under conditions in which resonant acoustic waves are excited in the tube at the other end. If the edge of the tube is round, separation does not occur at high Strouhal numbers, which enables us to verify theoretical predictions for dissipation in the boundary layer and for acoustic radiation. Observation with the aid of schlieren pictures shows that in the case of a sharp edge vortices are formed during inflow. The vortices are shed from the pipe during outflow. Based on these observations a mathematical model is developed for the generation and shedding of vorticity. The main result of the analysis is a boundary condition for the pressure in the wave, to be applied near the mouth. The pressure amplitudes in the acoustic wave measured under resonance are compared with theoretical predictions made with the aid of the boundary condition obtained in the paper.

scholarly journals A Modified PML Acoustic Wave Equation

Symmetry ◽  
2019 ◽  
Vol 11 (2) ◽  
pp. 177 ◽  
Author(s):  
Dojin Kim
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Finite Difference Method ◽  
Acoustic Wave ◽  
Difference Method ◽  
Acoustic Waves ◽  
Auxiliary Variable ◽  
Acoustic Wave Equation ◽  
Von Neumann ◽  
Standard Galerkin Method

In this paper, we consider a two-dimensional acoustic wave equation in an unbounded domain and introduce a modified model of the classical un-split perfectly matched layer (PML). We apply a regularization technique to a lower order regularity term employed in the auxiliary variable in the classical PML model. In addition, we propose a staggered finite difference method for discretizing the regularized system. The regularized system and numerical solution are analyzed in terms of the well-posedness and stability with the standard Galerkin method and von Neumann stability analysis, respectively. In particular, the existence and uniqueness of the solution for the regularized system are proved and the Courant-Friedrichs-Lewy (CFL) condition of the staggered finite difference method is determined. To support the theoretical results, we demonstrate a non-reflection property of acoustic waves in the layers.

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