Hostname: page-component-7dd5485656-wjfd9 Total loading time: 0 Render date: 2025-10-30T18:52:19.523Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  27 March 2012

Ludovic Moya
Ludovic Moya*
Affiliation:
INRIA Sophia Antipolis - Méditerannée, NACHOS project-team, 2004 Route des Lucioles, BP 93, 06902 Sophia Antipolis Cedex, France. ludovic.moya@inria.fr
Get access

Abstract

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

Keywords

Temporal convergencediscontinuous Galerkin methodtime-domain Maxwell equationscomponent splittingorder reduction

Information

Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 5 , September 2012 , pp. 1225 - 1246
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Références

Botchev, M.A. and Verwer, J.G., Numerical integration of damped maxwell equations. SIAM J. Sci. Comput. 31 (2009) 1322–1346. Google Scholar
Buffa, A. and Perugia, I., Discontinuous Galerkin approximation of the Maxwell eigenproblem. SIAM J. Numer. Anal. 44 (2006) 2198–2226. Google Scholar
Catella, A., Dolean, V. and Lanteri, S., 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. Google Scholar
B. Cockburn, G.E.G.E. Karniadakis and C.-W. Shu Eds., Discontinuous Galerkin methods. Theory, computation and applications. Springer-Verlag, Berlin (2000)
Cohen, G., Ferrieres, X. and Pernet, S., A spatial high order hexahedral discontinuous Galerkin method to solve Maxwell’s equations in time-domain. J. Comput. Phys. 217 (2006) 340–363. Google Scholar
Diaz, J. and Grote, M.J., Energy conserving explicit local time-stepping for second-order wave equations. SIAM J. Sci. Comput. 31 (2009) 1985–2014. Google Scholar
Dolean, V., Fahs, H., Fezoui, L. and Lanteri, S., Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. 229 (2010) 512–526. Google Scholar
Fahs, H., 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. Google Scholar
Faragó, I., Havasi, Á. and Zlatev, Z., Richardson-extrapolated sequential splitting and its application. J. Comput. Appl. Math. 234 (2010) 3283–3302. Google Scholar
Fezoui, L., Lanteri, S., Lohrengel, S. and Piperno, S., 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. Google Scholar
Grote, M.J. and Mitkova, T., Explicit local time stepping methods for Maxwell’s equations. J. Comput. Appl. Math. 234 (2010) 3283–3302. Google Scholar
E. Hairer and G. Wanner, Solving Ordinary Differential Equations II – Stiff and Differential-Algebraic problems, 2nd edition. Springer-Verlag, Berlin (1996).
E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration, 2nd edition. Springer-Verlag, Berlin (2002).
Hesthaven, J. and Warburton, T., Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181 (2002) 186–221. Google Scholar
J. Hesthaven and T. Warburton, Nodal Discontinuous Galerkin Methods. Springer (2008).
W. Hundsdorfer and J.G. Verwer, Numerical Solution of Time-Dependent Advection-Diffusion-Reaction Equations. Springer-Verlag, Berlin (2003).
J. Jin, The Finite Element Method in Electromagnetics, 2nd edition. Wiley-IEEE Press (2002).
Kulikov, G.Yu., Local theory of extrapolation methods. Numer. Algorithm 53 (2010) 321-342 Google Scholar
McLachlan, R.I., On the numerical integration of ordinary differential equations by symmetric composition methods. SIAM J. Sci. Comput. 16 (1995) 151–168. Google Scholar
Montseny, E., Pernet, S., Ferrires, X. and Cohen, G., 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. Google Scholar
Nédélec, J.C., Mixed finite elements in R3. Numer. Math. 35 (1980) 315–341. Google Scholar
Nédélec, J.C., A new dfamily of mixed finite elements in R3. Numer. Math. 50 (1986) 57–81. Google Scholar
Piperno, S., Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problem. ESAIM : M2AN 40 (2006) 815–841. Google Scholar
Remaki, M., A new finite volume scheme for solving Maxwell’s system. Compel 19 (2000) 913-931. Google Scholar
Suzuki, M., Fractal decomposition of exponential operators with applications to many-body theories and Monte-Carlo simulations. Phys. Lett. A 146 (1990) 319–323. Google Scholar
Taube, A., Dumbser, M., Munz, C.D. and Schneider, R., A high order discontinuous Galerkin method with local time stepping for the Maxwell equations. Int. J. Numer. Model. 22 (2009) 77–103. Google Scholar
Verwer, J.G., Component splitting for semi-discrete Maxwell equations. BIT Numer. Math. 51 (2011) 427–445.Google Scholar
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.
Verwer, J.G. and Botchev, M.A., Unconditionaly stable integration of Maxwell’s equations. Linear Algebra Appl. 431 (2009) 300–317. Google Scholar
Verwer, J.G. and de Vries, H.B., Global extrapolation of a first order splitting method. SIAM J. Sci. Stat. Comput. 6 (1985) 771–780. Google Scholar
Yee, K.S., Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antennas Propag. 14 (1966) 302–307. Google Scholar
Yoshida, H., Construction of higher order symplectic integrators. Phys. Lett. A 150 (1990) 262–268. CrossRefGoogle Scholar