Hochbruck, Marlis; Pažur, Tomislav; Schulz, Andreas; Thawinan, Ekkachai; Wieners, Christian Efficient time integration for discontinuous Galerkin approximations of linear wave equations [Plenary lecture presented at the 83rd annual GAMM conference, Darmstadt, 26th – 30th March, 2012]. (English) Zbl 1322.65095 ZAMM, Z. Angew. Math. Mech. 95, No. 3, 237-259 (2015). Summary: We consider the combination of discontinuous Galerkin discretizations in space with various time integration methods for linear acoustic, elastic, and electro-magnetic wave equations. For the discontinuous Galerkin method we derive explicit formulas for the full upwind flux for heterogeneous materials by solving the Riemann problems for the corresponding first-order systems. In a framework of bounded semigroups we prove convergence of the spatial discretization.For the time integration we discuss advantages and disadvantages of explicit and implicit Runge-Kutta methods compared to polynomial and rational Krylov subspace methods for the approximation of the matrix exponential function. Finally, the efficiency of the different time integrators is illustrated by several examples in 2D and 3D for electro-magnetic and elastic waves. Cited in 22 Documents MSC: 65M60 Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs 35L45 Initial value problems for first-order hyperbolic systems 35L65 Hyperbolic conservation laws 74J05 Linear waves in solid mechanics 76Q05 Hydro- and aero-acoustics 78M25 Numerical methods in optics (MSC2010) Keywords:wave equation; discontinuous Galerkin method; upwind flux; implicit Runge-Kutta method; exponential integrators; Krylov subspace method; matrix exponential function PDFBibTeX XMLCite \textit{M. Hochbruck} et al., ZAMM, Z. Angew. Math. Mech. 95, No. 3, 237--259 (2015; Zbl 1322.65095) Full Text: DOI References: [1] Al-Mohy, SIAM J. Sci. Comp. 33(2) pp 488– (2011) · Zbl 1234.65028 [2] Botchev, SIAM J. Sci. Comput. 35(3) (2013) [3] Botchev, J. Comput. Phys. 216(2) pp 654– (2006) · Zbl 1136.78328 [4] Burman, SIAM J. Numer. Anal. 48(6) pp 2019– (2010) · Zbl 1226.65086 [5] Chen, J. Sci. Comput. 22(1-3) pp 205– (2005) · Zbl 1091.78015 [6] D.A. Di Pietro A. Ern [7] Druskin, SIAM J. Sci. Comput. 35(2) (2013) [8] Eiermann, SIAM J. Matrix Anal. Appl. 32(2) pp 621– (2011) · Zbl 1264.65070 [9] K.J. Engel R. Nagel [10] Fezoui, M2AN Math. Model. Numer. Anal. 39(6) pp 1149– (2005) · Zbl 1094.78008 [11] Gallopoulos, SIAM J. Sci. Statist. Comput. 13(5) pp 1236– (1992) · Zbl 0757.65101 [12] Göckler, SIAM J. Numer. Anal. 51(4) pp 2189– (2013) · Zbl 1278.65076 [13] Gottlieb, SIAM Rev. 43(1) pp 89– (2001) · Zbl 0967.65098 [14] Grimm, BIT Numer. Math. 52(3) pp 639– (2012) · Zbl 1258.65052 [15] Grote, J. Comput. Appl. Math. 234(12) pp 3283– (2010) · Zbl 1210.78026 [16] Güttel, GAMM-Mitteilungen 36(1) pp 8– (2013) · Zbl 1292.65043 [17] E. Hairer G. Wanner [18] J.S. Hesthaven T. Warburton [19] Hesthaven, J. Comput. Phys. 181(1) pp 186– (2002) · Zbl 1014.78016 [20] N.J. Higham [21] Hochbruck, SIAM J. Numer. Anal. 34(5) pp 1911– (1997) · Zbl 0888.65032 [22] M. Hochbruck T. Pažur [23] Jetschny, Geophys. Prospect. 58(2) pp 245– (2010) [24] Käser, Geophys. J. Int. 166 pp 855– (2006) [25] Klöckner, J. Comput. Phys. 228(21) pp 7863– (2009) · Zbl 1175.65111 [26] Krivodonova, J. Comput. Phys. 229(22) pp 8537– (2010) · Zbl 1201.65171 [27] R.J. Leveque [28] Luo, J. Comput. Phys. 211(2) pp 767– (2006) · Zbl 1138.76408 [29] Maurer, Parallel Comput. 37 pp 742– (2011) · Zbl 06094856 [30] Remis, IEEE Trans. Microw. Theory Tech. 45(12) pp 2139– (1997) [31] M. Renardy R.C. Rogers [32] Saad, SIAM J. Numer. Anal. 29(1) pp 209– (1992) · Zbl 0749.65030 [33] Sármány, J. Sci. Comput. 33(1) pp 47– (2007) · Zbl 1128.78014 [34] R. Shirazi-Nejad [35] Sidje, ACM Trans. Math. Softw. 24(1) pp 130– (1998) · Zbl 0917.65063 [36] Taube, Int. J. Numer. Model., Electron. Netw. Devices Fields (UK) 22(1) pp 77– (2009) · Zbl 1156.78012 [37] van den Eshof, SIAM J. Sci. Comput. 27(4) pp 1438– (2006) · Zbl 1105.65051 [38] Vila, Numer. Math. 94(3) pp 573– (2003) · Zbl 1030.65110 [39] Wieners, Comput. Visual. Sci. 13 pp 161– (2010) · Zbl 1216.65164 [40] C. Wieners [41] Zaslavsky, J. Comput. Phys. 229(12) pp 4831– (2010) · Zbl 1205.65273 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.