Stability of the Staggered-Grid High-Order Difference Method for FIRST-Order Elastic Wave Equation

2000 ◽  
Vol 43 (6) ◽  
pp. 904-913 ◽  
Author(s):  
Liang-Guo DONG ◽  
Zai-Tian MA ◽  
Jing-Zhong CAO
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Difference Method ◽  
High Order ◽  
Staggered Grid ◽  
Elastic Wave Equation ◽  
Order Difference ◽  
First Order

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.

Staggered-Grid High-Order Differencemethod for the First-Order Elastic Wave Equations

2000 ◽  
Vol 43 (3) ◽  
pp. 441-449 ◽  
Author(s):  
Liang-Guo DONG ◽  
Zai-Tian MA ◽  
Jing-Zhong CAO
Keyword(s):  
Elastic Wave ◽  
Wave Equations ◽  
High Order ◽  
Staggered Grid ◽  
First Order ◽  
Elastic Wave Equations

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

Elastic wave equation modeling using staggered‐grid convolutional differentiators

1994 ◽  
Author(s):  
Binhong Zhou ◽  
Iian M. Mason ◽  
Stewart A. Greenhalgh
Keyword(s):  
Wave Equation ◽  
Elastic Wave ◽  
Staggered Grid ◽  
Equation Modeling ◽  
Elastic Wave Equation
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