Energy conservative SBP discretizations of the acoustic wave equation in covariant form on staggered curvilinear grids. (English) Zbl 1436.65108
Summary: We develop a numerical method for solving the acoustic wave equation in covariant form on staggered curvilinear grids in an energy conserving manner. The use of a covariant basis decomposition leads to a rotationally invariant scheme that outperforms a Cartesian basis decomposition on rotated grids. The discretization is based on high order Summation-By-Parts (SBP) operators and preserves both symmetry and positive definiteness of the contravariant metric tensor. To improve accuracy and decrease computational cost, we also derive a modified discretization of the metric tensor that leads to a conditionally stable discretization. Bounds are derived that yield a point-wise condition that can be evaluated to check for stability of the modified discretization. This condition shows that the interpolation operators should be constructed such that their norm is close to one.
MSC:
| 65M06 | Finite difference methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
summation-by-parts; high order accuracy; staggered grids; curvilinear coordinates; covariant formulation; acoustic wave equationReferences:
| [1] | Carpenter, M. H.; Gottlieb, D.; Arbabanel, S.; Don, W.-S., The theoretical accuracy of Runge-Kutta time discretizations for the initial boundary value problem: a study of the boundary error, SIAM J. Sci. Comput., 16, 1241-1252 (1995) · Zbl 0839.65098 |
| [2] | de la Puente, Josep; Ferrer, Miguel; Hanzich, Mauricio; Castillo, José E.; Cela, José M., Mimetic seismic wave modeling including topography on deformed staggered grids, Geophysics, 79, 3, T125-T141 (2014) |
| [3] | Gao, Longfei; Del Rey Fernández, David C.; Carpenter, Mark; Keyes, David, Sbp-sat finite difference discretization of acoustic wave equations on staggered block-wise uniform grids, J. Comput. Appl. Math., 348, 421-444 (2019) · Zbl 1412.65075 |
| [4] | Gao, Longfei; Keyes, David, Combining finite element and finite difference methods for isotropic elastic wave simulations in an energy-conserving manner, J. Comput. Phys., 378, 665-685 (2019) · Zbl 1416.65263 |
| [5] | Gassner, Gregor J., A skew-symmetric discontinuous galerkin spectral element discretization and its relation to sbp-sat finite difference methods, SIAM J. Sci. Comput., 35, 3, A1233-A1253 (2013) · Zbl 1275.65065 |
| [6] | Grinfeld, Pavel, Introduction to Tensor Analysis and the Calculus of Moving Surfaces (2013), Springer · Zbl 1300.53001 |
| [7] | Gustafsson, B., The convergence rate for difference approximations to mixed initial boundary value problems, Math. Comput., 29, 130, 396-406 (1975) · Zbl 0313.65085 |
| [8] | Hestholm, Stig, Elastic wave modeling with free surfaces: stability of long simulations, Geophysics, 68, 1, 314-321 (2003) |
| [9] | Hestholm, Stig; Ruud, Bent, 2d finite-difference elastic wave modelling including surface topography, Geophys. Prospect., 42, 5, 371-390 (1994) |
| [10] | Hestholm, Stig; Ruud, Bent, 3-d finite-difference elastic wave modeling including surface topography, Geophysics, 63, 2, 613-622 (1998) |
| [11] | Hestholm, Stig; Ruud, Bent, 3d free-boundary conditions for coordinate-transform finite-difference seismic modelling, Geophys. Prospect., 50, 5, 463-474 (2002) |
| [12] | Kozdon, Jeremy E.; Wilcox, Lucas C., Stable coupling of nonconforming, high-order finite difference methods, SIAM J. Sci. Comput., 38, 2, A923-A952 (2016) · Zbl 1380.65160 |
| [13] | 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) · Zbl 0355.65085 |
| [14] | Lundquist, Tomas; Malan, Arnaud; Nordström, Jan, A hybrid framework for coupling arbitrary summation-by-parts schemes on general meshes, J. Comput. Phys., 362, 49-68 (2018) · Zbl 1391.76560 |
| [15] | Mattsson, Ken, Diagonal-norm summation by parts operators for finite difference approximations of third and fourth derivatives, J. Comput. Phys., 274, 432-454 (2014) · Zbl 1352.65249 |
| [16] | Mattsson, Ken; Carpenter, Mark 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 |
| [17] | Mattsson, Ken, Ossian O’Reilly. Compatible diagonal-norm staggered and upwind sbp operators, J. Comput. Phys. (2017) · Zbl 1380.65166 |
| [18] | Nordström, Jan; Forsberg, Karl; Adamsson, Carl; Eliasson, Peter, Finite volume methods, unstructured meshes and strict stability for hyperbolic problems, Appl. Numer. Math., 45, 4, 453-473 (2003) · Zbl 1019.65066 |
| [19] | Nordström, Jan; Gong, Jing, A stable hybrid method for hyperbolic problems, J. Comput. Phys., 212, 2, 436-453 (2006) · Zbl 1083.65085 |
| [20] | Olsson, P., Summation by parts, projections, and stability. I, Math. Comput., 64, 1035-1065 (1995) · Zbl 0828.65111 |
| [21] | Olsson, P., Summation by parts, projections, and stability. II, Math. Comput., 64, 1473-1493 (1995) · Zbl 0848.65064 |
| [22] | O’Reilly, Ossian; Lundquist, Tomas; Dunham, Eric M.; Nordström, Jan, Energy stable and high-order-accurate finite difference methods on staggered grids, J. Comput. Phys., 346, 572-589 (2017) · Zbl 1380.65173 |
| [23] | Pérez Solano, C. A.; Donno, D.; Chauris, H., Finite-difference strategy for elastic wave modelling on curved staggered grids, Comput. Geosci., 20, 1, 245-264 (Feb. 2016) · Zbl 1392.86024 |
| [24] | Petersson, N. A.; O’Reilly, O.; Sjögreen, B.; Bydlon, S., Discretizing singular point sources in hyperbolic wave propagation problems, J. Comput. Phys., 321, 532-555 (2016) · Zbl 1349.65329 |
| [25] | Petersson, N. A.; Sjögreen, B., Stable grid refinement and singular source discretization for seismic wave simulations, Commun. Comput. Phys., 8, 1074-1110 (2010) · Zbl 1364.86010 |
| [26] | Petersson, N. A.; Sjögreen, B., Wave propagation in anisotropic elastic materials and curvilinear coordinates using a summation-by-parts finite difference method, J. Comput. Phys., 299, 820-841 (2015) · Zbl 1351.74159 |
| [27] | Petersson, N. Anders; Sjögreen, Björn, High order accurate finite difference modeling of seismo-acoustic wave propagation in a moving atmosphere and a heterogeneous earth model coupled across a realistic topography, J. Sci. Comput., 74, 1, 290-323 (Jan. 2018) · Zbl 1404.65100 |
| [28] | Prochnow, Bo; O’Reilly, Ossian; Dunham, Eric M.; Petersson, N. Anders, 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 |
| [29] | Ranocha, Hendrik; Öffner, Philipp; Sonar, Thomas, Summation-by-parts operators for correction procedure via reconstruction, J. Comput. Phys., 311, 299-328 (2016) · Zbl 1349.65524 |
| [30] | Roache, Patrick J., Code verification by the method of manufactured solutions, J. Fluids Eng., 124, 1, 4-10 (2002) |
| [31] | Sjögreen, B.; Petersson, N. A., A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation, J. Sci. Comput., 52, 17-48 (2012) · Zbl 1255.65162 |
| [32] | Strand, B., Summation by parts for finite difference approximations for d/dx, J. Comput. Phys., 110, 47-67 (1994) · Zbl 0792.65011 |
| [33] | Taflove, Allen; Hagness, Susan C., Computational Electrodynamics: the Finite-Difference Time-Domain Method (2005), Artech House · Zbl 0963.78001 |
| [34] | Tarrass, Issam; Giraud, Luc; Thore, Pierre, New curvilinear scheme for elastic wave propagation in presence of curved topography, Geophys. Prospect., 59, 5, 889-906 (2011) |
| [35] | Thompson, J. F.; Warsi, Z. U.A.; Mastin, C. W., Numerical Grid Generation: Foundations and Applications (1985), Elsevier North-Holland, Inc.: Elsevier North-Holland, Inc. New York, NY, USA · Zbl 0598.65086 |
| [36] | Wang, Siyang; Kreiss, Gunilla, Convergence of summation-by-parts finite difference methods for the wave equation, J. Sci. Comput., 71, 1, 219-245 (Apr. 2017) · Zbl 1398.65222 |
| [37] | Xie, Zhongqiang; Chan, Chi-Hou; Zhang, Bo, An explicit fourth-order orthogonal curvilinear staggered-grid fdtd method for maxwell’s equations, J. Comput. Phys., 175, 2, 739-763 (2002) · Zbl 1009.78008 |
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