Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations
Moya, Ludovic
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012), p. 1225-1246 / Harvested from Numdam
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In this paper we study the temporal convergence of a locally implicit discontinuous Galerkin method for the time-domain Maxwell's equations modeling electromagnetic waves propagation. Particularly, we wonder whether the method retains its second-order ordinary differential equation (ODE) convergence under stable simultaneous space-time grid refinement towards the true partial differential equation (PDE) solution. This is not a priori clear due to the component splitting which can introduce order reduction

Publié le : 2012-01-01
DOI : https://doi.org/10.1051/m2an/2012002
Classification:  65M12,  65M60,  78M10 
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     author = {Moya, Ludovic},
     title = {Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     volume = {46},
     year = {2012},
     pages = {1225-1246},
     doi = {10.1051/m2an/2012002},
     mrnumber = {2916379},
     zbl = {1277.78036},
     language = {en},
     url = {http://dml.mathdoc.fr/item/M2AN_2012__46_5_1225_0}
}

Moya, Ludovic. Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) pp. 1225-1246. doi : 10.1051/m2an/2012002. http://gdmltest.u-ga.fr/item/M2AN_2012__46_5_1225_0/

[1] M.A. Botchev and J.G. Verwer, Numerical integration of damped maxwell equations. SIAM J. Sci. Comput. 31 (2009) 1322-1346. | MR 2486832 | Zbl 1229.78026

[2] A. Buffa and I. Perugia, Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44 (2006) 2198-2226. | MR 2263045 | Zbl pre05202318

[3] A. Catella, V. Dolean and S. Lanteri, An unconditionally stable discontinuous galerkin method for solving the 2-D time-domain Maxwell equations on unstructured triangular meshes. IEEE Trans. Magn. 44 (2008) 1250-1253.

[4] B. Cockburn, G.E.G.E. Karniadakis and C.-W. Shu Eds., Discontinuous Galerkin methods. Theory, computation and applications. Springer-Verlag, Berlin (2000) | MR 1842160 | Zbl 0989.76045

[5] G. Cohen, X. Ferrieres and S. Pernet, A spatial high order hexahedral discontinuous Galerkin method to solve Maxwell's equations in time-domain. J. Comput. Phys. 217 (2006) 340-363. | MR 2260605 | Zbl 1160.78004

[6] J. Diaz and M.J. Grote, Energy conserving explicit local time-stepping for second-order wave equations. SIAM J. Sci. Comput. 31 (2009) 1985-2014. | MR 2516141 | Zbl 1195.65131

[7] V. Dolean, H. Fahs, L. Fezoui and S. Lanteri, Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. 229 (2010) 512-526. | MR 2565614 | Zbl 1213.78037

[8] H. Fahs, Development of a hp-like discontinuous Galerkin time-domain method on non-conforming simplicial meshes for electromagnetic wave propagation. Int. J. Numer. Anal. Mod. 6 (2009) 193-216. | MR 2574904 | Zbl 1158.78329

[9] I. Faragó, Á. Havasi and Z. Zlatev, Richardson-extrapolated sequential splitting and its application. J. Comput. Appl. Math. 234 (2010) 3283-3302. | Zbl 1160.65022

[10] L. Fezoui, S. Lanteri, S. Lohrengel and S. Piperno, Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM : M2AN 39 (2005) 1149-1176. | Numdam | MR 2195908 | Zbl 1094.78008

[11] M.J. Grote and T. Mitkova, Explicit local time stepping methods for Maxwell's equations. J. Comput. Appl. Math. 234 (2010) 3283-3302. | MR 2665386 | Zbl 1210.78026

[12] E. Hairer and G. Wanner, Solving Ordinary Differential Equations II - Stiff and Differential-Algebraic problems, 2nd edition. Springer-Verlag, Berlin (1996). | MR 1439506 | Zbl 0729.65051

[13] E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, 2nd edition. Springer-Verlag, Berlin (2002). | MR 1904823 | Zbl 0994.65135

[14] J. Hesthaven and T. Warburton, Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell's equations. J. Comput. Phys. 181 (2002) 186-221. | MR 1925981 | Zbl 1014.78016

[15] J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. Springer (2008). | MR 2372235 | Zbl 1134.65068

[16] W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003). | MR 2002152 | Zbl 1030.65100

[17] J. Jin, The Finite Element Method in Electromagnetics, 2nd edition. Wiley-IEEE Press (2002). | MR 1903357 | Zbl 0823.65124

[18] G.Yu. Kulikov, Local theory of extrapolation methods. Numer. Algorithm 53 (2010) 321-342 | MR 2600933 | Zbl 1186.65089

[19] R.I. Mclachlan, On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16 (1995) 151-168. | MR 1311683 | Zbl 0821.65048

[20] E. Montseny, S. Pernet, X. Ferrires and G. Cohen, Dissipative terms and local time-stepping improvements in a spatial high order Discontinuous Galerkin scheme for the time-domain Maxwell's equations. J. Comput. Phys. 227 (2008) 6795-6820. | MR 2435431 | Zbl 1144.78330

[21] J.C. Nédélec, Mixed finite elements in R3. Numer. Math. 35 (1980) 315-341. | Zbl 0419.65069

[22] J.C. Nédélec, A new dfamily of mixed finite elements in R3. Numer. Math. 50 (1986) 57-81. | Zbl 0625.65107

[23] S. Piperno, Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problem. ESAIM : M2AN 40 (2006) 815-841. | Numdam | MR 2293248 | Zbl 1121.78014

[24] M. Remaki, A new finite volume scheme for solving Maxwell's system. Compel 19 (2000) 913-931. | Zbl 0994.78021

[25] M. Suzuki, Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations. Phys. Lett. A 146 (1990) 319-323. | MR 1059400

[26] A. Taube, M. Dumbser, C.D. Munz and R. Schneider, A high order discontinuous Galerkin method with local time stepping for the Maxwell equations. Int. J. Numer. Model. 22 (2009) 77-103. | Zbl 1156.78012

[27] J.G. Verwer, Component splitting for semi-discrete Maxwell equations. BIT Numer. Math. 51 (2011) 427-445. | MR 2806538 | Zbl 1221.65247

[28] J.G Verwer, Composition methods, Maxwell's and source term. CWI Technical report (2010); Available at http://oai.cwi.nl/oai/asset/17036/17036A.pdf.

[29] J.G. Verwer and M.A. Botchev, Unconditionaly stable integration of Maxwell's equations. Linear Algebra Appl. 431 (2009) 300-317. | MR 2528933 | Zbl 1170.78004

[30] J.G. Verwer and H.B. De Vries, Global extrapolation of a first order splitting method. SIAM J. Sci. Stat. Comput. 6 (1985) 771-780. | MR 791198 | Zbl 0592.65060

[31] K.S. Yee, Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propag. 14 (1966) 302-307. | Zbl 1155.78304

[32] H. Yoshida, Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262-268. | MR 1078768