Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions
Abstract
:1. Introduction
2. Related Work
3. Methodology and Formulation
3.1. First-Order Scheme
3.2. Second-Order Scheme
3.3. The Stability Criterion
4. Numerical Procedure and Results
5. Conclusions
Author Contributions
Funding
Conflicts of Interest
Appendix A
Appendix A.1. The Truncation Errors of the Scheme (4)/(6)/(7)
Appendix A.2. The Truncation Errors of the Scheme (9)/(10)/(11)
Appendix B
Appendix B.1. The Proof of Theorem 2
Appendix B.2. The Proof of Theorem 3
Appendix C
Symbol | Parameter | Symbol | Parameter |
---|---|---|---|
v | solid’s velocity | w | fluid’s velocity |
the total stress of the bulk material | P | the total fluid pressure | |
porosity | M | the uniaxial modulus of the skeleton | |
poroelastic coefficient | viscosity | ||
permeability | F | the Biot-flow coefficient | |
S | the characteristic squirt-flow coefficient | solid’s density | |
fluid’s density | the additional coupling density | ||
f | frequency | the eigenvalue of the matrix | |
the time-step size | the spatial increment |
Acronym | Full-Form | Acronym | Full-Form |
---|---|---|---|
FD method | finite-difference method | BISQ model | Biot/squirt model |
FE method | finite-element method | CFL condition | Courant–Friedrichs–Lewy condition |
3D | three-dimension |
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Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|
M (GPa) | 29 | (%) | 15 | 0.5702 | |
(kg/m) | 2600 | (kg/m) | 1000 | (kg/m) | 450 |
(kg/ms) | 100 | f (KHz) | 50 | R (m) | 0.001 |
N | Order | Order | Order | |||
---|---|---|---|---|---|---|
80 | 3.48 × | − | 1.36 | − | 7.08 × | − |
160 | 1.86 × | 0.90 | 7.93 × | 0.77 | 3.86 × | 0.88 |
320 | 9.71 × | 0.94 | 4.23 × | 0.91 | 2.16 × | 0.83 |
640 | 4.43 × | 1.13 | 2.08 × | 1.02 | 1.13 × | 0.93 |
1280 | 2.0 3× | 1.12 | 1.02 × | 1.03 | 5.80 × | 0.97 |
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Liu, J.; Yong, W.-A.; Liu, J.; Guo, Z. Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions. Mathematics 2020, 8, 1039. https://doi.org/10.3390/math8061039
Liu J, Yong W-A, Liu J, Guo Z. Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions. Mathematics. 2020; 8(6):1039. https://doi.org/10.3390/math8061039
Chicago/Turabian StyleLiu, Jiawei, Wen-An Yong, Jianxin Liu, and Zhenwei Guo. 2020. "Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions" Mathematics 8, no. 6: 1039. https://doi.org/10.3390/math8061039
APA StyleLiu, J., Yong, W.-A., Liu, J., & Guo, Z. (2020). Stable Finite-Difference Methods for Elastic Wave Modeling with Characteristic Boundary Conditions. Mathematics, 8(6), 1039. https://doi.org/10.3390/math8061039