scholarly journals Stability of High-Order Staggered-Grid Schemes for 3D Elastic Wave Equation in Heterogeneous Media

2019 ◽  
Vol 07 (08) ◽  
pp. 1755-1774
Author(s):  
Atish Kumar Joardar ◽  
Wensheng Zhang
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Heterogeneous Media ◽  
High Order ◽  
Staggered Grid ◽  
Elastic Wave Equation

scholarly journals Simulation of elastic wave propagation across fractures using a nodal discontinuous Galerkin method—theory, implementation and validation

2019 ◽  
Vol 219 (3) ◽  
pp. 1900-1914 ◽  
Author(s):  
T Möller ◽  
W Friederich
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Elastic Wave ◽  
Discontinuous Galerkin ◽  
Heterogeneous Media ◽  
Rheological Behaviour ◽  
Displacement Discontinuity ◽  
Signal Frequency ◽  
Elastic Wave Equation ◽  
Numerical Flux

SUMMARY An existing nodal discontinuous Galerkin (NDG) method for the simulation of seismic waves in heterogeneous media is extended to media containing fractures with various rheological behaviour. Fractures are treated as 2-D surfaces where Schoenberg’s linear slip or displacement discontinuity condition is applied as an additional boundary condition to the elastic wave equation which is in turn implemented as an additional numerical flux within the NDG formulation. Explicit expressions for the new numerical flux are derived by considering the Riemann problem for the elastic wave equation at fractures with varying rheologies. In all cases, we obtain further first order differential equations that fully describe the temporal evolution of the particle velocity jump at the fracture. Our flux formulation allows to separate the effect of a fracture from flux contributions due to simple welded interfaces enabling us to easily declare element faces as parts of a fracture. We make use of this fact by first generating the numerical mesh and then building up fractures by selecting appropriate element faces instead of adjusting the mesh to pre-defined fracture surfaces. The implementation of the new numerical fluxes into NDG is verified in 1-D by comparison to an analytical solution and in 2-D by comparing the results of a simulation valid in 1-D and 2-D. Further numerical examples address the effect of fracture systems on seismic wave propagation in 1-D and 2-D featuring effective anisotropy and coda generation. Finally, a study of the reflective and transmissive behaviour of fractures indicates that reflection and transmission coefficients are controlled by the ratio of signal frequency and relaxation frequency of the fracture.

Temporal high-order staggered-grid finite-difference schemes for elastic wave propagation

Geophysics ◽  
2017 ◽  
Vol 82 (5) ◽  
pp. T207-T224 ◽  
Author(s):  
Zhiming Ren ◽  
Zhen Chun Li
Keyword(s):  
Wave Propagation ◽  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Propagation ◽  
Staggered Grid ◽  
S Wave ◽  
Order Accuracy ◽  
Spatial Derivatives

The traditional high-order finite-difference (FD) methods approximate the spatial derivatives to arbitrary even-order accuracy, whereas the time discretization is still of second-order accuracy. Temporal high-order FD methods can improve the accuracy in time greatly. However, the present methods are designed mainly based on the acoustic wave equation instead of elastic approximation. We have developed two temporal high-order staggered-grid FD (SFD) schemes for modeling elastic wave propagation. A new stencil containing the points on the axis and a few off-axial points is introduced to approximate the spatial derivatives. We derive the dispersion relations of the elastic wave equation based on the new stencil, and we estimate FD coefficients by the Taylor series expansion (TE). The TE-based scheme can achieve ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracy ([Formula: see text]). We further optimize the coefficients of FD operators using a combination of TE and least squares (LS). The FD coefficients at the off-axial and axial points are computed by TE and LS, respectively. To obtain accurate P-, S-, and converted waves, we extend the wavefield decomposition into the temporal high-order SFD schemes. In our modeling, P- and S-wave separation is implemented and P- and S-wavefields are propagated by P- and S-wave dispersion-relation-based FD operators, respectively. We compare our schemes with the conventional SFD method. Numerical examples demonstrate that our TE-based and TE + LS-based schemes have greater accuracy in time and better stability than the conventional method. Moreover, the TE + LS-based scheme is superior to the TE-based scheme in suppressing the spatial dispersion. Owing to the high accuracy in the time and space domains, our new SFD schemes allow for larger time steps and shorter operator lengths, which can improve the computational efficiency.

scholarly journals High-order cut finite elements for the elastic wave equation

2020 ◽  
Vol 46 (3) ◽  
Author(s):  
Simon Sticko ◽  
Gustav Ludvigsson ◽  
Gunilla Kreiss
Keyword(s):  
Wave Equation ◽  
Finite Elements ◽  
Elastic Wave ◽  
High Order ◽  
Elastic Wave Equation

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.

scholarly journals Finite-Difference Simulation of Elastic Wave with Separation in Pure P- and S-Modes

2014 ◽  
Vol 2014 ◽  
pp. 1-14 ◽  
Author(s):  
Ke-Yang Chen
Keyword(s):  
Wave Equation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Computing Time ◽  
Staggered Grid ◽  
Difference Operators ◽  
Elastic Wave Equation ◽  
Absorbing Boundary ◽  
Wave Separation ◽  
Time Requirements

Elastic wave equation simulation offers a way to study the wave propagation when creating seismic data. We implement an equivalent dual elastic wave separation equation to simulate the velocity, pressure, divergence, and curl fields in pure P- and S-modes, and apply it in full elastic wave numerical simulation. We give the complete derivations of explicit high-order staggered-grid finite-difference operators, stability condition, dispersion relation, and perfectly matched layer (PML) absorbing boundary condition, and present the resulting discretized formulas for the proposed elastic wave equation. The final numerical results of pure P- and S-modes are completely separated. Storage and computing time requirements are strongly reduced compared to the previous works. Numerical testing is used further to demonstrate the performance of the presented method.

Time-space-domain temporal high-order staggered-grid finite-difference schemes by combining orthogonality and pyramid stencils for 3D elastic-wave propagation

Geophysics ◽  
2019 ◽  
Vol 84 (4) ◽  
pp. T259-T282 ◽  
Author(s):  
Shigang Xu ◽  
Yang Liu ◽  
Zhiming Ren ◽  
Hongyu Zhou
Keyword(s):  
Dispersion Relation ◽  
Finite Difference ◽  
Elastic Wave ◽  
Large Time ◽  
High Order ◽  
Staggered Grid ◽  
Elastic Wave Equation ◽  
Space Domain ◽  
First Order ◽  
Time Space

The presently available staggered-grid finite-difference (SGFD) schemes for the 3D first-order elastic-wave equation can only achieve high-order spatial accuracy, but they exhibit second-order temporal accuracy. Therefore, the commonly used SGFD methods may suffer from visible temporal dispersion and even instability when relatively large time steps are involved. To increase the temporal accuracy and stability, we have developed a novel time-space-domain high-order SGFD stencil, characterized by ([Formula: see text])th-order spatial and ([Formula: see text])th-order temporal accuracies ([Formula: see text]), to numerically solve the 3D first-order elastic-wave equation. The core idea of this new stencil is to use a double-pyramid stencil with an operator length parameter [Formula: see text] together with the conventional second-order SGFD to approximate the temporal derivatives. At the same time, the spatial derivatives are discretized by the orthogonality stencil with an operator length parameter [Formula: see text]. We derive the time-space-domain dispersion relation of this new stencil and determine finite-difference (FD) coefficients using the Taylor-series expansion. In addition, we further optimize the spatial FD coefficients by using a least-squares (LS) algorithm to minimize the time-space-domain dispersion relation. To create accurate and reasonable P-, S-, and converted wavefields, we introduce the 3D wavefield-separation technique into our temporal high-order SGFD schemes. The decoupled P- and S-wavefields are extrapolated by using the P- and S-wave dispersion-relation-based FD coefficients, respectively. Moreover, we design an adaptive variable-length operator scheme, including operators [Formula: see text] and [Formula: see text], to reduce the extra computational cost arising from adopting this new stencil. Dispersion and stability analyses indicate that our new methods have higher accuracy and better stability than the conventional ones. Using several 3D modeling examples, we demonstrate that our SGFD schemes can yield greater temporal accuracy on the premise of guaranteeing high-order spatial accuracy. Through effectively combining our new stencil, LS-based optimization, large time step, variable-length operator, and graphic processing unit, the computational efficiency can be significantly improved for the 3D case.

Sign in / Sign up

Export Citation Format

Share Document