Journal of Scientific Computing

, Volume 56, Issue 1, pp 190–218 | Cite as

Locally Implicit Time Integration Strategies in a Discontinuous Galerkin Method for Maxwell’s Equations

  • Stéphane Descombes
  • Stéphane Lanteri
  • Ludovic Moya


An attractive feature of discontinuous Galerkin (DG) spatial discretization is the possibility of using locally refined space grids to handle geometrical details. However, locally refined meshes lead to severe stability constraints on explicit integration methods to numerically solve a time-dependent partial differential equation. If the region of refinement is small relative to the computational domain, the time step size restriction can be overcome by blending an implicit and an explicit scheme where only the solution variables living at fine elements are treated implicitly. The downside of this approach is having to solve a linear system per time step. But due to the assumed small region of refinement relative to the computational domain, the overhead will also be small while the solution can be advanced in time with step sizes determined by the coarse elements. In this paper, we present two locally implicit time integration methods for solving the time-domain Maxwell equations spatially discretized with a DG method. Numerical experiments for two-dimensional problems illustrate the theory and the usefulness of the implicit–explicit approaches in presence of local refinements.


Discontinuous Galerkin spatial discretization  Locally implicit time integration methods Time-domain Maxwell equations 


  1. 1.
    Ascher, U.M., Ruuth, S.J., Spiteri, R.J.: Implicit–explicit Runge-Kutta methods for time-dependent partial differential equations. Appl. Numer. Math. 25(2), 151–167 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Botchev, M.A., Verwer, J.G.: Numerical integration of damped Maxwell equations. SIAM J. SCI. Comput. 31(2), 1322–1346 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cai, W., Deng, S.: An upwinding embedded boundary method for Maxwell’s equations in media with material interfaces: 2D case. J. Comput. Phys. 190(1), 159–183 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Calvo, M.P., de Frutos, J., Novo, J.: Linearly implicit Runge-Kutta methods for advection reaction diffusion equations. Appl. Numer. Math. 37(4), 535–549 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Cockburn, B., Karniadakis, G.E., Shu, C.W.: Discontinuous Galerkin methods. Theory, Computation and Applications. Springer, Berlin (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Crouzeix, M.: Une méthode multipas implicite-explicite pour l’approximation des équations d’évolution paraboliques. Numer. Math. 35(3), 257–276 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Diaz, J., Grote, M.: Energy conserving explicit local time-stepping for second-order wave equations. SIAM J. SCI. Comput. 31(3), 1985–2014 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Dolean, V., Fahs, H., Fezoui, L., Lanteri, S.: Locally implicit discontinuous Galerkin method for time domain electromagnetics. J. Comput. Phys. 229(2), 512–526 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    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(2), 193–216 (2009)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Fezoui, L., Lanteri, S., Lohrengel, S., Piperno, S.: Convergence and stability of a discontinuous Galerkin time-domain method for the 3D heterogeneous Maxwell equations on unstructured meshes. ESAIM: M2AN 39(6), 1149–1176 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Grote, M.J., Mitkova, T.: Explicit local time stepping methods for Maxwell’s equations. J. Comp. Appl. Math. 234(12), 3283–3302 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Hesthaven, J.S., Warburton, T.: Nodal high-order methods on unstructured grids. I. Time-domain solution of Maxwell’s equations. J. Comput. Phys. 181(1), 186–221 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods. Springer, Berlin (2008)CrossRefzbMATHGoogle Scholar
  14. 14.
    Kanevsky, A., Carpenter, M.H., Gottlieb, D., Hesthaven, J.S.: Application of implicit-explicit high order Runge-Kutta methods to discontinuous-Galerkin schemes. J. Comput. Phys. 225(2), 1753–1781 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kennedy, C.A., Carpenter, M.H.: Additive RungeKutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44(1–2), 139–181 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Moya, L.: Temporal convergence of a locally implicit discontinuous Galerkin method for Maxwell’s equations. ESAIM: M2AN 46(5), 1225–1246 (2012)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Piperno, S.: Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problem. ESAIM: M2AN 40(5), 815–841 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Reed, W.H., Hill, T.R.: Triangular mesh methods for the neutron transport equation. Los Alamos Scientific Laboratory, Report LAUR-73-479 (1973)Google Scholar
  19. 19.
    Sármány, D., Botchev, M.A., Van der Vegt, J.J.W.: Dispersion and dissipation error in high-order Runge-Kutta discontinuous Galerkin discretisations of the Maxwell equations. J. Sci. Comput. 33(1), 47–74 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Taube, A., Dumbser, M., Munz, C.D., Schneider, R.: A high order discontinuous Galerkin method with local time stepping for the Maxwell equations. Int. J. Numer. Model. Elec. Netw. Dev. Fields 22(1), 77–103 (2009)CrossRefzbMATHGoogle Scholar
  21. 21.
    Varah, J.M.: Stability restrictions on second, three- level finite-difference schemes for parabolic equations. SIAM J. Numer. Anal. 17(2), 300–309 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Verwer, J.G.: Component splitting for semi-discrete Maxwell equations. BIT Numer. Math. 51(2), 427–445 (2010)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Verwer, J.G., Botchev, M.A.: Unconditionaly stable integration of Maxwell’s equations. Linear Algebra Appl. 431(3–4), 300–317 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Wesseling, P.: Principles of Computational Fluid Dynamics. Springer Series in Computational Mathematics, vol. 29. Berlin (2001)Google Scholar
  25. 25.
    Yee, K.S.: Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media. IEEE Trans. Antenn. Propag. 14(3), 302–307 (1966)Google Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Stéphane Descombes
    • 1
  • Stéphane Lanteri
    • 2
  • Ludovic Moya
    • 2
  1. 1.J.A. Dieudonné Mathematics Laboratory, CNRS UMR 7351University Nice Sophia AntipolisNice CedexFrance
  2. 2.Inria Sophia Antipolis—Méditerranée, Nachos project-team 2004 Route des LuciolesSophia Antipolis CedexFrance

Personalised recommendations