An unsplit complex frequency-shifted perfectly matched layer for second-order acoustic wave equations

2021 ◽  
Author(s):  
Xiuzheng Fang ◽  
Fenglin Niu
Keyword(s):  
Acoustic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Complex Frequency

Nonsplit complex-frequency shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 1: Method

Geophysics ◽  
2018 ◽  
Vol 83 (6) ◽  
pp. T301-T311 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xueyuan Huang ◽  
Yanjie Zhou
Keyword(s):  
Boundary Condition ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Absorbing Boundary Conditions ◽  
Second Order ◽  
Complex Frequency ◽  
Absorbing Boundary ◽  
Time Layer ◽  
The Time Domain

The absorbing boundary condition plays an important role in seismic wave modeling. The perfectly matched layer (PML) boundary condition has been established as one of the most effective and prevalent absorbing boundary conditions. Among the existing PML-type conditions, the complex frequency shift (CFS) PML attracts considerable attention because it can handle the evanescent and grazing waves better. For solving the resultant CFS-PML equation in the time domain, one effective technique is to apply convolution operations, which forms the so-called convolutional PML (CPML). We have developed the corresponding CPML conditions with nonconstant grid compression parameter, and used its combination algorithms specifically with the symplectic partitioned Runge-Kutta and the nearly analytic SPRK methods for solving second-order seismic wave equations. This involves evaluating second-order spatial derivatives with respect to the complex stretching coordinates at the noninteger time layer. Meanwhile, two kinds of simplification algorithms are proposed to compute the composite convolutions terms contained therein.

A stable auxiliary differential equation perfectly matched layer condition combined with low-dispersive symplectic methods for solving second-order elastic wave equations

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T193-T206 ◽  
Author(s):  
Xiao Ma ◽  
Yangjia Li ◽  
Jiaxing Song
Keyword(s):  
Differential Equation ◽  
Elastic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Elastic Model ◽  
Complex Frequency ◽  
Symplectic Method ◽  
Symplectic Methods ◽  
Compression Parameter

The stable implementation of the perfectly matched layer (PML), one of the most effective and popular artificial boundary conditions, has attracted much attention these years. As a type of low-dispersive and symplectic method for solving seismic wave equations, the nearly-analytic symplectic partitioned Runge-Kutta (NSPRK) method has been combined with split-field PML (SPML) and convolutional complex-frequency shifted PML (C-CFS-PML) previously to model acoustic and short-time elastic wave modelings, not yet successfully applied to long-time elastic wave propagation. In order to broaden the application of NSPRK and more general symplectic methods for second-order seismic models, we formulate an auxiliary differential equation (ADE)-CFS-PML with a stabilizing grid compression parameter. This includes deriving the ADE-CFS-PML equations and formulating an adequate time integrator to properly embed their numerical discretizations in the main symplectic numerical methods. The resulting (N)SPRK+ADE-CFS-PML algorithm can help break through the constraint of at most second-order temporal accuracy that used to be imposed on SPML and C-CFS-PML. Especially for NSPRK, we implement the strategy of neglecting the treatment of third-order spatial derivatives in the PML domain and obtain an efficient absorption effect. Related acoustic and elastic wave simulations illustrate the enhanced numerical accuracy of our ADE-CFS-PML compared with SPML and C-CFS-PML. The elastic wave simulation in a homogeneous isotropic medium shows that compared to NSPRK+C-CFS-PML, the NSPRK+ADE-CFS-PML is numerically stable throughout a simulation time of 2 s. The synthetic seismograms of the 2D acoustic SEG salt model and the two-layer elastic model demonstrate the effectiveness of NSPRK+ADE-CFS-PML for complex elastic models. The stabilization effect of the grid compression parameter is verified in the final homogeneous isotropic elastic model with free-surface boundary.

scholarly journals A Well-Posed and Discretely Stable Perfectly Matched Layer for Elastic Wave Equations in Second Order Formulation

2012 ◽  
Vol 11 (5) ◽  
pp. 1643-1672 ◽  
Author(s):  
Kenneth Duru ◽  
Gunilla Kreiss
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Order System ◽  
Stable Node ◽  
Complex Frequency ◽  
Elastic Wave Equations ◽  
Anisotropic Elastic ◽  
Well Posed

AbstractWe present a well-posed and discretely stable perfectly matched layer for the anisotropic (and isotropic) elastic wave equations without first re-writing the governing equations as a first order system. The new model is derived by the complex coordinate stretching technique. Using standard perturbation methods we show that complex frequency shift together with a chosen real scaling factor ensures the decay of eigen-modes for all relevant frequencies. To buttress the stability properties and the robustness of the proposed model, numerical experiments are presented for anisotropic elastic wave equations. The model is approximated with a stable node-centered finite difference scheme that is second order accurate both in time and space.

Nonsplit complex-frequency-shifted perfectly matched layer combined with symplectic methods for solving second-order seismic wave equations — Part 2: Wavefield simulations

Geophysics ◽  
2019 ◽  
Vol 84 (3) ◽  
pp. T167-T179 ◽  
Author(s):  
Xiao Ma ◽  
Dinghui Yang ◽  
Xijun He ◽  
Xueyuan Huang ◽  
Jiaxing Song
Keyword(s):  
Boundary Condition ◽  
Wave Equation ◽  
Free Surface ◽  
Seismic Wave ◽  
Wave Equations ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Surface Boundary ◽  
Complex Frequency ◽  
Symplectic Methods

The perfectly matched layer (PML) is an efficient artificial boundary condition that has been routinely implemented in seismic wave modeling. However, the effective combination of PML with symplectic numerical schemes for solving seismic wave equations has rarely been studied. In a companion paper, we have developed a complex-frequency-shifted convolutional PML (CPML) with a nonconstant compression grid parameter for solving the time-domain second-order seismic wave equation. Subsequently, we combine this CPML with two classes of symplectic methods to formulate symplectic partitioned Runge-Kutta (SPRK) + CPML and nearly analytic SPRK (NSPRK) + CPML, both of which are properly synchronized. To further investigate their validity, the two algorithms are then applied to acoustic and elastic wave simulations in typical geologic models, including a heterogeneous acoustic model, several isotropic and orthotropic elastic models, and an isotropic elastic model with a free-surface boundary. Relevant numerical results demonstrate the effectiveness of our CPML and combination algorithms. Specifically, the numerical accuracy and stability of the CPML that we develop are greatly improved compared with the classic split-field PML. Moreover, the final model with the free-surface boundary condition indicates that the nonconstant grid-compression parameter can eliminate the unstable modes at the free surface in the PML domain. The (N)SPRK + CPML that we propose is prospective for future application in other complex models and wave-equation-based migration and inversion.

Time-space-domain implicit finite-difference methods for modeling acoustic wave equations

Geophysics ◽  
2018 ◽  
Vol 83 (4) ◽  
pp. T175-T193 ◽  
Author(s):  
Enjiang Wang ◽  
Jing Ba ◽  
Yang Liu
Keyword(s):  
Finite Difference ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Finite Difference Methods ◽  
Second Order ◽  
Computational Time ◽  
Time Step ◽  
Space Domain ◽  
Time Space ◽  
Temporal Dispersion

It has been proved that the implicit spatial finite-difference (FD) method can obtain higher accuracy than explicit FD by using an even smaller operator length. However, when only second-order FD in time is used, the combined FD scheme is prone to temporal dispersion and easily becomes unstable when a relatively large time step is used. The time-space domain FD can suppress the temporal dispersion. However, because the spatial derivatives are solved explicitly, the method suffers from spatial dispersion and a large spatial operator length has to be adopted. We have developed two effective time-space-domain implicit FD methods for modeling 2D and 3D acoustic wave equations. First, the high-order FD is incorporated into the discretization for the second-order temporal derivative, and it is combined with the implicit spatial FD. The plane-wave analysis method is used to derive the time-space-domain dispersion relations, and two novel methods are proposed to determine the spatial and temporal FD coefficients in the joint time-space domain. First, we fix the implicit spatial FD coefficients and derive the quadratic convex objective function with respect to temporal FD coefficients. The optimal temporal FD coefficients are obtained by using the linear least-squares method. After obtaining the temporal FD coefficients, the SolvOpt nonlinear algorithm is applied to solve the nonquadratic optimization problem and obtain the optimized temporal and spatial FD coefficients simultaneously. The dispersion analysis, stability analysis, and modeling examples validate that the proposed schemes effectively increase the modeling accuracy and improve the stability conditions of the traditional implicit schemes. The computational efficiency is increased because the schemes can adopt larger time steps with little loss of spatial accuracy. To reduce the memory requirement and computational time for storing and calculating the FD coefficients, we have developed the representative velocity strategy, which only computes and stores the FD coefficients at several selected velocities. The modeling result of the 2D complicated model proves that the representative velocity strategy effectively reduces the memory requirements and computational time without decreasing the accuracy significantly when a proper velocity interval is used.

Perfectly matched layer on curvilinear grid for the second-order seismic acoustic wave equation

2014 ◽  
Vol 45 (2) ◽  
pp. 94-104 ◽  
Author(s):  
Sanyi Yuan ◽  
Shangxu Wang ◽  
Wenju Sun ◽  
Lina Miao ◽  
Zhenhua Li
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Acoustic Wave Equation ◽  
Curvilinear Grid

A novel unsplit perfectly matched layer for the second-order acoustic wave equation

Ultrasonics ◽  
2014 ◽  
Vol 54 (6) ◽  
pp. 1568-1574 ◽  
Author(s):  
Youneng Ma ◽  
Jinhua Yu ◽  
Yuanyuan Wang
Keyword(s):  
Wave Equation ◽  
Acoustic Wave ◽  
Perfectly Matched Layer ◽  
Second Order ◽  
Acoustic Wave Equation

scholarly journals RECOVERING THE GRADIENTS OF THE SOLUTIONS OF SECOND ORDER HYPERBOLIC EQUATIONS BY INTERPOLATING THE FINITE ELEMENT SOLUTIONS

1995 ◽  
Vol 03 (01) ◽  
pp. 27-56 ◽  
Author(s):  
TAO LIN ◽  
HONG WANG
Keyword(s):  
Finite Element ◽  
Acoustic Wave ◽  
Wave Equations ◽  
Polynomial Interpolation ◽  
Hyperbolic Equations ◽  
Second Order ◽  
Acoustic Wave Equation ◽  
Degree Polynomial ◽  
First Order ◽  
Derivatives Of

We present a technique to generate better approximations to the gradients of the solutions of the second order hyperbolic (acoustic wave) equations by postprocessing finite element solutions with higher degree polynomial interpolation. The postprocessing procedure is inexpensive, local, vectorizable, and parallelizable. In addition, the postprocessing procedure is independent of the computation of the finite element solution; therefore it can be easily incorporated with the existing finite element codes to efficiently generate globally better approximations to the gradients or first order partial derivatives of the main quantities of the acoustic field modeled by the acoustic wave equation.

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