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Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems

Published online by Cambridge University Press:  16 January 2007

Serge Piperno
Serge Piperno*
Affiliation:
Cermics, project-team caiman, École des Ponts, ParisTech, INRIA, France.
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Abstract

The Discontinuous Galerkin Time Domain (DGTD) methods are now popular for the solution of wave propagation problems. Able to deal with unstructured, possibly locally-refined meshes, they handleeasily complex geometries and remain fully explicit with easy parallelization and extension to high orders of accuracy. Non-dissipative versions exist, where some discrete electromagnetic energy is exactly conserved. However, the stability limit of the methods, related to the smallest elements in the mesh, calls for the construction of local-time stepping algorithms. These schemes have already been developed for N-body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.

Keywords

WavesacousticsMaxwell's systemDiscontinuous Galerkin methodssymplectic schemesenergy conservationsecond-order accuracy.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 40 , Issue 5 , September 2006 , pp. 815 - 841
Copyright
© EDP Sciences, SMAI, 2007

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References

Bécache, E., Joly, P. and Rodríguez, J., Space-time mesh refinement for elastodynamics. Numerical results. Comput. Method. Appl. M. 194 (2005) 355–366. CrossRef
N. Canouet, L. Fezoui and S. Piperno, A new Discontinuous Galerkin method for 3D Maxwell's equations on non-conforming grids, in Proc. Sixth International Conference on Mathematical and Numerical Aspects of Wave Propagation, G.C. Cohen et al. Ed., Springer, Jyväskylä, Finland (2003) 389–394.
C. Chauviere, J.S. Hesthaven, A. Kanevsky and T. Warburton, High-order localized time integration for grid-induced stiffness, in Proc. Second M.I.T. Conference on Computational Fluid and Solid Mechanics, Cambridge, MA (2003).
J.-P. Cioni, L. Fezoui, L. Anne and F. Poupaud, A parallel FVTD Maxwell solver using 3D unstructured meshes, in Proc. 13th annual review of progress in applied computational electromagnetics, Monterey, California (1997) 359–365.
B. Cockburn, G.E. Karniadakis, C.-W. Shu Eds., Discontinuous Galerkin methods. Theory, computation and applications 11 Lect. Notes Comput. Sci. Engrg., Springer-Verlag, Berlin (2000).
Cockburn, B. and Shu, C.-W., Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173–261. CrossRef
Collino, F., Fouquet, T. and Joly, P., Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations. J. Comput. Phys. 211 (2006) 9–35. CrossRef
Dawson, C. and Kirby, R., High resolution schemes for conservation laws with locally varying time steps. SIAM J. Sci. Comput. 22 (2001) 2256–2281. CrossRef
Elmkies, A. and Joly, P., Éléments finis d'arête et condensation de masse pour les équations de Maxwell: le cas de dimension 3. C. R. Acad. Sci. Paris Sér. I Math. 325 (1997) 1217–1222. CrossRef
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. CrossRef
Hardy, D.J., Okunbor, D.I. and Skeel, R.D., Symplectic variable step size integration for N-body problems. Appl. Numer. Math. 29 (1999) 19–30. CrossRef
Hesthaven, J. and Teng, C., Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21 (2000) 2352–2380. CrossRef
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. CrossRef
Hesthaven, J. and Warburton, T., High-order nodal discontinuous Galerkin methods for the maxwell eigenvalue problem. Philos. Trans. Roy. Soc. London Ser. A 362 (2004) 493–524. CrossRef
Hirono, T., Lui, W.W. and Yokoyama, K., Time-domain simulation of electromagnetic field using a symplectic integrator. IEEE Microwave Guided Wave Lett. 7 (1997) 279–281. CrossRef
Hirono, T., Lui, W.W., Yokoyama, K. and Seki, S., Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation. J. Lightwave Tech. 16 (1998) 1915–1920. CrossRef
Holder, T., Leimkuhler, B. and Reich, S., Explicit variable step-size and time-reversible integration. Appl. Numer. Math. 39 (2001) 367–377. CrossRef
Huang, W. and Leimkuhler, B., The adaptive Verlet method. SIAM J. Sci. Comput. 18 (1997) 239–256. CrossRef
Hyman, J.M. and Shashkov, M., Mimetic discretizations for Maxwell's equations. J. Comput. Phys. 151 (1999) 881–909. CrossRef
P. Joly and C. Poirier, A new second order 3D edge element on tetrahedra for time dependent Maxwell's equations, in Proc. Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation, A. Bermudez, D. Gomez, C. Hazard, P. Joly, J.-E. Roberts Eds., SIAM, Santiago de Compostella, Spain (2000) 842–847.
Kennedy, C.A. and Carpenter, M.H., Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44 (2003) 139–181. CrossRef
D.A. Kopriva, S.L. Woodruff and M.Y. Hussaini, Discontinuous spectral element approximation of Maxwell's equations, in Discontinuous Galerkin methods. Theory, computation and applications 11 Lect. Notes Comput. Sci. Engrg. B. Cockburn, G.E. Karniadakis, C.-W. Shu Eds., Springer-Verlag, Berlin (2000) 355–362.
Leimkuhler, B., Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories. Philos. Trans. Roy. Soc. London Ser. A 357 (1999) 1101–1134. CrossRef
Lu, X. and Schmid, R., Symplectic discretization for Maxwell's equations. J. Math. Computing 25 (2001) 1–21.
S. Piperno, Fully explicit DGTD methods for wave propagation on time-and-space locally refined grids, in Proc. Seventh International Conference on Mathematical and Numerical Aspects of Wave Propagation, Providence, RI (2005) 402–404.
J.-F. Remacle, K. Pinchedez, J.E. Flaherty and M.S. Shephard, An efficient local time stepping-discontinuous Galerkin scheme for adaptive transient computations. Technical report 2001-13, Rensselaer Polytechnic Institute (2001).
Remaki, M., A new finite volume scheme for solving Maxwell's system. COMPEL 19 (2000) 913–931. CrossRef
Rieben, R., White, D. and Rodrigue, G., High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations. IEEE Trans. Antennas Propagation 52 (2004) 2190–2195. CrossRef
J.M. Sanz-Serna and M.P. Calvo, Numerical Hamiltonian Problems, Chapman and Hall, London, UK (1994).
Shang, J. and Fithen, R., A comparative study of characteristic-based algorithms for the Maxwell equations. J. Comput. Phys. 125 (1996) 378–394. CrossRef
T. Warburton, Application of the discontinuous Galerkin method to Maxwell's equations using unstructured polymorphic hp-finite elements, in Discontinuous Galerkin methods. Theory, computation and applications 11 Lect. Notes Computat. Sci. Engrg., B. Cockburn, G.E. Karniadakis, C.-W. Shu Eds., Springer-Verlag, Berlin (2000) 451–458.
T. Warburton, Spurious solutions and the Discontinuous Galerkin method on non-conforming meshes, in Proc. Seventh International Conference on Mathematical and Numerical Aspects of Wave Propagation, Providence, RI (2005) 405–407.
Yee, K.S., Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propagation 16 (1966) 302–307.