SBP-SAT finite difference discretization of acoustic wave equations on staggered block-wise uniform grids. (English) Zbl 1412.65075
Summary: We consider the numerical simulation of the acoustic wave equations arising from seismic applications, for which staggered grid finite difference methods are popular choices due to their simplicity and efficiency. We relax the uniform grid restriction on finite difference methods and allow the grids to be block-wise uniform with nonconforming interfaces. In doing so, variations in the wave speeds of the subterranean media can be accounted for more efficiently. Staggered grid finite difference operators satisfying the summation-by-parts (SBP) property are devised to approximate the spatial derivatives appearing in the acoustic wave equation. These operators are applied within each block independently. The coupling between blocks is achieved through simultaneous approximation terms (SATs), which impose the interface conditions weakly, i.e., by penalty. Ratio of the grid spacing of neighboring blocks is allowed to be rational number, for which specially designed interpolation formulas are presented. These interpolation formulas constitute key pieces of the simultaneous approximation terms. The overall discretization is shown to be energy-conserving and examined on test cases of both theoretical and practical interests, delivering accurate and stable simulation results.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
| 86A15 | Seismology (including tsunami modeling), earthquakes |
| 76Q05 | Hydro- and aero-acoustics |
| 65M50 | Mesh generation, refinement, and adaptive methods for the numerical solution of initial value and initial-boundary value problems involving PDEs |
| 35Q86 | PDEs in connection with geophysics |
Keywords:
summation by parts; simultaneous approximation terms; seismic wave modeling; nonconforming interface; staggered grid; long time instabilityReferences:
| [1] | Tarantola, A., Inversion of seismic reflection data in the acoustic approximation, Geophysics, 49, 8, 1259-1266, (1984) |
| [2] | Luo, Y.; Schuster, G. T., Wave-equation traveltime inversion, Geophysics, 56, 5, 645-653, (1991) |
| [3] | Symes, W. W., Migration velocity analysis and waveform inversion, Geophys. Prospect., 56, 6, 765-790, (2008) |
| [4] | Virieux, J.; Operto, S., An overview of full-waveform inversion in exploration geophysics, Geophysics, 74, 6, WCC1-WCC26, (2009) |
| [5] | Komatitsch, D.; Vilotte, J. P., The spectral element method: An efficient tool to simulate the seismic response of 2D and 3D geological structures, Bull. Seismol. Soc. Am., 88, 2, 368-392, (1998) · Zbl 0974.74583 |
| [6] | Komatitsch, D.; Tromp, J., Introduction to the spectral element method for three-dimensional seismic wave propagation, Geophys. J. Int., 139, 3, 806-822, (1999) |
| [7] | Käser, M.; Dumbser, M., An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes -I. The two-dimensional isotropic case with external source terms, Geophys. J. Int., 166, 2, 855-877, (2006) |
| [8] | Virieux, J., SH-wave propagation in heterogeneous media: Velocity-stress finite-difference method, Geophysics, 49, 11, 1933-1942, (1984) |
| [9] | Levander, A. R., Fourth-order finite-difference P-SV seismograms, Geophysics, 53, 11, 1425-1436, (1988) |
| [10] | Saenger, E. H.; Bohlen, T., Finite-difference modeling of viscoelastic and anisotropic wave propagation using the rotated staggered grid, Geophysics, 69, 2, 583-591, (2004) |
| [11] | Hayashi, K.; Burns, D. R.; Toksöz, M. N., Discontinuous-grid finite-difference seismic modeling including surface topography, Bull. Seismol. Soc. Am., 91, 6, 1750-1764, (2001) |
| [12] | Kristek, J.; Moczo, P.; Galis, M., Stable discontinuous staggered grid in the finite-difference modelling of seismic motion, Geophys. J. Int., 183, 3, 1401-1407, (2010) |
| [13] | Zhang, Z.; Zhang, W.; Li, H.; Chen, X., Stable discontinuous grid implementation for collocated-grid finite-difference seismic wave modelling, Geophys. J. Int., 192, 3, 1179-1188, (2013) |
| [14] | Kreiss, H. O.; Scherer, G., Finite element and finite difference methods for hyperbolic partial differential equations, (Mathematical Aspects of Finite Elements in Partial Differential Equations, (1974), Academic Press), 195-212 · Zbl 0355.65085 |
| [15] | Carpenter, M. H.; Gottlieb, D.; Abarbanel, S., Time-stable boundary conditions for finite-difference schemes solving hyperbolic systems: methodology and application to high-order compact schemes, J. Comput. Phys., 111, 2, 220-236, (1994) · Zbl 0832.65098 |
| [16] | Del Rey Fernández, D. C.; Hicken, J. E.; Zingg, D. W., Review of summation-by-parts operators with simultaneous approximation terms for the numerical solution of partial differential equations, Comput. & Fluids, 95, 171-196, (2014) · Zbl 1390.65064 |
| [17] | Svärd, M.; Nordström, J., Review of summation-by-parts schemes for initial-boundary-value problems, J. Comput. Phys., 268, 17-38, (2014) · Zbl 1349.65336 |
| [18] | Gao, L.; Ketcheson, D.; Keyes, D., On long-time instabilities in staggered finite difference simulations of the seismic acoustic wave equations on discontinuous grids, Geophys. J. Int., 212, 2, 1098-1110, (2018) |
| [19] | Gao, L.; Brossier, R.; Virieux, J., Using time filtering to control the long-time instability in seismic wave simulation, Geophys. J. Int., 204, 3, 1443-1461, (2016) |
| [20] | O’Reilly, O.; Lundquist, T.; Dunham, E. M.; Nordström, J., Energy stable and high-order-accurate finite difference methods on staggered grids, J. Comput. Phys., 346, 572-589, (2017) · Zbl 1380.65173 |
| [21] | Prochnow, B.; O’Reilly, O.; Dunham, E. M.; Petersson, N. A., Treatment of the polar coordinate singularity in axisymmetric wave propagation using high-order summation-by-parts operators on a staggered grid, Comput. & Fluids, 149, 138-149, (2017) · Zbl 1390.76613 |
| [22] | Mattsson, K.; O’Reilly, O., Compatible diagonal-norm staggered and upwind SBP operators, J. Comput. Phys., 352, 52-75, (2018) · Zbl 1375.76111 |
| [23] | Hicken, J. E.; Zingg, D. W., Summation-by-parts operators and high-order quadrature, J. Comput. Appl. Math., 237, 1, 111-125, (2013) · Zbl 1263.65025 |
| [24] | Mattsson, K.; Carpenter, M. H., Stable and accurate interpolation operators for high-order multiblock finite difference methods, SIAM J. Sci. Comput., 32, 4, 2298-2320, (2010) · Zbl 1216.65107 |
| [25] | Mattsson, K.; Almquist, M.; van der Weide, E., Boundary optimized diagonal-norm SBP operators, J. Comput. Phys., (2018) · Zbl 1416.65275 |
| [26] | Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110, 1, 47-67, (1994) · Zbl 0792.65011 |
| [27] | Mattsson, K., Boundary procedures for summation-by-parts operators, J. Sci. Comput., 18, 1, 133-153, (2003) · Zbl 1024.76031 |
| [28] | Bodony, D. J., Accuracy of the simultaneous-approximation-term boundary condition for time-dependent problems, J. Sci. Comput., 43, 1, 118-133, (2010) · Zbl 1203.76043 |
| [29] | Steeb, W. H.; Shi, T. K., Matrix Calculus and Kronecker Product with Applications and C++ Programs, (1997), World Scientific · Zbl 0952.15019 |
| [30] | Van Loan, C. F., The ubiquitous Kronecker product, J. Comput. Appl. Math., 123, 1, 85-100, (2000) · Zbl 0966.65039 |
| [31] | Wang, S.; Virta, K.; Kreiss, G., High order finite difference methods for the wave equation with non-conforming grid interfaces, J. Sci. Comput., 68, 3, 1002-1028, (2016) · Zbl 1352.65274 |
| [32] | Versteeg, R., The Marmousi experience: Velocity model determination on a synthetic complex data set, Lead. Edge, 13, 9, 927-936, (1994) |
| [33] | Svärd, M.; Mattsson, K.; Nordström, J., Steady-state computations using summation-by-parts operators, J. Sci. Comput., 24, 1, 79-95, (2005) · Zbl 1080.76044 |
| [34] | Diener, P.; Dorband, E. N.; Schnetter, E.; Tiglio, M., Optimized high-order derivative and dissipation operators satisfying summation by parts, and applications in three-dimensional multi-block evolutions, J. Sci. Comput., 32, 1, 109-145, (2007) · Zbl 1120.65092 |
| [35] | Petersson, N. A.; Sjogreen, B., Stable grid refinement and singular source discretization for seismic wave simulations, Commun. Comput. Phys., 8, 5, 1074-1110, (2010) · Zbl 1364.86010 |
| [36] | Berenger, J. P., A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114, 2, 185-200, (1994) · Zbl 0814.65129 |
| [37] | Komatitsch, D.; Martin, R., An unsplit convolutional perfectly matched layer improved at grazing incidence for the seismic wave equation, Geophysics, 72, 5, SM155-SM167, (2007) |
| [38] | Duru, K.; Kozdon, J. E.; Kreiss, G., Boundary conditions and stability of a perfectly matched layer for the elastic wave equation in first order form, J. Comput. Phys., 303, 372-395, (2015) · Zbl 1349.74359 |
| [39] | Duru, K., The role of numerical boundary procedures in the stability of perfectly matched layers, SIAM J. Sci. Comput., 38, 2, A1171-A1194, (2016) · Zbl 1339.65112 |
| [40] | Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comp., 29, 130, 396-406, (1975) · Zbl 0313.65085 |
| [41] | Gustafsson, B., The convergence rate for difference approximations to general mixed initial-boundary value problems, SIAM J. Numer. Anal., 18, 2, 179-190, (1981) · Zbl 0469.65068 |
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