# zbMATH — the first resource for mathematics

A fourth order accurate finite difference scheme for the elastic wave equation in second order formulation. (English) Zbl 1255.65162
Summary: We present a fourth order accurate finite difference method for the elastic wave equation in second order formulation, where the fourth order accuracy holds in both space and time. The key ingredient of the method is a boundary modified fourth order accurate discretization of the second derivative with variable coefficient, $$(\mu (x)u _{x })_{x }$$. This discretization satisfies a summation by parts identity that guarantees stability of the scheme. The boundary conditions are enforced through ghost points, thereby avoiding projections or penalty terms, which often are used with previous summation by parts operators. The temporal discretization is obtained by an explicit modified equation method. Numerical examples with free surface boundary conditions show that the scheme is stable for CFL-numbers up to 1.3, and demonstrate a significant improvement in efficiency over the second order accurate method. The new discretization of $$(\mu (x)u _{x })_{x }$$ has general applicability, and will enable stable fourth order accurate approximations of other partial differential equations as well as the elastic wave equation.

##### MSC:
 65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
WPP
Full Text:
##### References:
 [1] Appelö, D., Petersson, N.A.: A stable finite difference method for the elastic wave equation on complex geometries with free surfaces. Commun. Comput. Phys. 5, 84–107 (2009) · Zbl 1364.74016 [2] 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, 220–236 (1994) · Zbl 0832.65098 [3] Carpenter, M.H., Nordström, J., Gottlieb, D.: A stable and conservative interface treatment of arbitrary spatial accuracy. J. Comput. Phys. 148, 341–365 (1999) · Zbl 0921.65059 [4] Dumbser, M., Käser, M.: An arbitrary high-order discontinuous Galerkin method for elastic waves on unstructured meshes. II. The three-dimensional isotropic case. Geophys. J. Int. 167, 319–336 (2006) [5] Feng, K.-A., Teng, C.-H., Chen, M.-H.: A pseudospectral penalty scheme for 2D isotropic elastic wave computations. J. Sci. Comput. 33, 313–348 (2007) · Zbl 1127.74048 [6] Graves, R.W.: Simulating seismic wave propagation in 3D elastic media using staggered-grid finite differences. Bull. Seismol. Soc. Am. 86, 1091–1106 (1996) [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] Gustafsson, B., Kreiss, H.-O., Oliger, J.: Time Dependent Problems and Difference Methods. Wiley-Interscience, New York (1995) · Zbl 0843.65061 [9] Komatitsch, K., Tromp, J.: Introduction to the spectral element method for three-dimensional seismic wave propagation. Geophys. J. Int. 139, 806–822 (1999) [10] Kreiss, H.-O., Oliger, J.: Comparison of accurate methods for the integration of hyperbolic equations. Tellus 24, 199–215 (1972) [11] Kreiss, H.-O., Petersson, N.A.: Boundary estimates for the elastic wave equation in almost incompressible materials. SIAM J. Numer. Anal. (2011, submitted), LLNL-JRNL-482152 [12] Kreiss, H.-O., Scherer, G.: Finite element and finite difference methods for hyperbolic partial differential equations. In: Mathematical Aspects of Finite Elements in Partial Differential Equations. Academic Press, San Diego (1974) · Zbl 0355.65085 [13] Levander, A.R.: Fourth-order finite-difference P-SV seismograms. Geophysics 53, 1425–1436 (1988) [14] Mattson, K.: Summation-by-parts operators for high order finite difference methods. PhD thesis, Uppsala University, Information Technology, Department of Scientific Computing (2003) [15] Mattsson, K., Svärd, M., Shoeybi, M.: Stable and accurate schemes for the compressible Navier-Stokes equations. J. Comput. Phys. 227, 2293–2316 (2008) · Zbl 1132.76039 [16] Mattsson, K., Norström, J.: Summation by parts operators for finite difference approximations of second derivatives. J. Comput. Phys. 199, 503–540 (2004) · Zbl 1071.65025 [17] Nilsson, S., Petersson, N.A., Sjögreen, B., Kreiss, H.-O.: Stable difference approximations for the elastic wave equation in second order formulation. SIAM J. Numer. Anal. 45, 1902–1936 (2007) · Zbl 1158.65064 [18] Petersson, N.A., Sjögreen, B.: An energy absorbing far-field boundary condition for the elastic wave equation. Commun. Comput. Phys. 6, 483–508 (2009) · Zbl 1364.74019 [19] Petersson, N.A., Sjögreen, B.: Stable and efficient modeling of anelastic attenuation in seismic wave propagation. Technical Report LLNL-JRNL-460239, Lawrence Livermore National Laboratory, Commun. Comput. Phys. (2010, to appear) · Zbl 1373.74099 [20] 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 [21] Petersson, N.A., Sjögreen, B.: User’s guide to WPP version 2.1. Technical Report LLNL-SM-487431, Lawrence Livermore National Laboratory (2011). (Source code available from https://computation.llnl.gov/casc/serpentine ) [22] Sjögreen, B.: High order centered difference methods for the compressible Navier-Stokes equations. J. Comput. Phys. 117, 67–78 (1995) · Zbl 0817.76047 [23] Strand, B.: Summation by parts for finite difference approximations for d/dx. J. Comput. Phys. 110, 47–67 (1994) · Zbl 0792.65011 [24] Virieux, J.: P-SV wave propagation in heterogeneous media: velocity-stress finite-difference method. Geophysics 51, 889–901 (1986) [25] Wilcox, L.C., Stadler, G., Burstedde, C., Ghattas, O.: A high-order discontinuous Galerkin method for wave propagation through coupled elastic-acoustic media. J. Comput. Phys. 229, 9373–9396 (2010) · Zbl 1427.74071
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.