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.

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.

scholarly journals Stable Symmetric Matrix Form Framework for the Elastic Wave Equation Combined with Perfectly Matched Layer and Discretized in the Curve Domain

Symmetry ◽  
2020 ◽  
Vol 12 (2) ◽  
pp. 202
Author(s):  
Cheng Sun ◽  
Zailin Yang ◽  
Guanxixi Jiang
Keyword(s):  
Differential Equation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Symmetric Matrix ◽  
Perfectly Matched Layer ◽  
High Order ◽  
Matrix Form ◽  
Discrete Approximation ◽  
Complex Frequency ◽  
Elastic Wave Equation

In this paper, we present a stable and accurate high-order methodology for the symmetric matrix form (SMF) of the elastic wave equation. We use an accurate high-order upwind finite difference method to define spatial discretization. Then, an efficient complex frequency-shifted (CFS) unsplit multi-axis perfectly matched layer (MPML) is implemented using the auxiliary differential equation (ADE) that is used to build higher-order time schemes for elastodynamics in the unbounded curve domain. It is derived to be compatible with SMF. The SMF framework has a general form of a hyperbolic partial differential equation (PDE) that can be expanded to different dimensions (2D, 3D) or different wave modal (SH, P-SV) without requiring significant modifications owing to a simplified process of derivation and programming. Subsequently, an energy analysis on the framework combined with initial boundary value problems is conducted, and the stability analysis can be extended to a semi-discrete approximation similarly. Thus, we propose a semi-discrete approximation based on ADE CFS-MPML in which the curve domain is discretized using the upwind summation-by-parts (SBP) operators, and where the boundary conditions are enforced weakly using the simultaneous approximation terms (SAT). The proposed method’s robustness and adequacy are illustrated by conducting several numerical simulations.

A simple implementation of PML for second-order elastic wave equations

2020 ◽  
Vol 246 ◽  
pp. 106867 ◽  
Author(s):  
Mingwei Zhuang ◽  
Qiwei Zhan ◽  
Jianyang Zhou ◽  
Zichao Guo ◽  
Na Liu ◽  
...  
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
Second Order ◽  
Elastic Wave Equations ◽  
Simple Implementation
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