Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell's equations

Moya, Ludovic

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1225-1246.

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

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

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

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

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

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

[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 | Zbl

[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 | Zbl

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

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

[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 | Zbl

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

[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

[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

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

[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 | Zbl

[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 | Zbl

[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

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