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 handle easily 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 -body mechanical problems and are known as symplectic schemes. They are applied here to DGTD methods on wave propagation problems.
@article{M2AN_2006__40_5_815_0,
author = {Piperno, Serge},
title = {Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
volume = {40},
year = {2006},
pages = {815-841},
doi = {10.1051/m2an:2006035},
mrnumber = {2293248},
zbl = {1121.78014},
language = {en},
url = {http://dml.mathdoc.fr/item/M2AN_2006__40_5_815_0}
}
Piperno, Serge. Symplectic local time-stepping in non-dissipative DGTD methods applied to wave propagation problems. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) pp. 815-841. doi : 10.1051/m2an:2006035. http://gdmltest.u-ga.fr/item/M2AN_2006__40_5_815_0/
[1] , and, Space-time mesh refinement for elastodynamics. Numerical results. Comput. Method. Appl. M. 194 (2005) 355-366. | Zbl 1095.74030
[2] , and, 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. | Zbl 1075.78002
[3] ,, and, 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).
[4] ,, and, 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.
[5] ,, Eds., Discontinuous Galerkin methods. Theory, computation and applications 11 Lect. Notes Comput. Sci. Engrg., Springer-Verlag, Berlin (2000). | MR 1842160 | Zbl 0935.00043
[6] and, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems. J. Sci. Comput. 16 (2001) 173-261. | Zbl 1065.76135
[7] , and, Conservative space-time mesh refinement methods for the FDTD solution of Maxwell's equations. J. Comput. Phys. 211 (2006) 9-35. | Zbl 1107.78015
[8] and, High resolution schemes for conservation laws with locally varying time steps. SIAM J. Sci. Comput. 22 (2001) 2256-2281. | Zbl 0980.35015
[9] and, 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. | Zbl 0893.65068
[10] ,, and, 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 | Zbl 1094.78008
[11] , and, Symplectic variable step size integration for -body problems. Appl. Numer. Math. 29 (1999) 19-30. | Zbl 0927.70003
[12] and, Stable spectral methods on tetrahedral elements. SIAM J. Sci. Comput. 21 (2000) 2352-2380. | Zbl 0959.65112
[13] and, Nodal high-order methods on unstructured grids. I: Time-domain solution of Maxwell's equations. J. Comput. Phys. 181 (2002) 186-221. | Zbl 1014.78016
[14] and, High-order nodal discontinuous Galerkin methods for the maxwell eigenvalue problem. Philos. Trans. Roy. Soc. London Ser. A 362 (2004) 493-524. | Zbl 1078.78014
[15] , and, Time-domain simulation of electromagnetic field using a symplectic integrator. IEEE Microwave Guided Wave Lett. 7 (1997) 279-281.
[16] ,, and, Stability and numerical dispersion of symplectic fourth-order time-domain schemes for optical field simulation. J. Lightwave Tech. 16 (1998) 1915-1920.
[17] , and, Explicit variable step-size and time-reversible integration. Appl. Numer. Math. 39 (2001) 367-377. | Zbl 0991.65060
[18] and, The adaptive Verlet method. SIAM J. Sci. Comput. 18 (1997) 239-256. | Zbl 0877.65048
[19] and, Mimetic discretizations for Maxwell's equations. J. Comput. Phys. 151 (1999) 881-909. | Zbl 0956.78015
[20] and, 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. | Zbl 0995.78040
[21] and, Additive Runge-Kutta schemes for convection-diffusion-reaction equations. Appl. Numer. Math. 44 (2003) 139-181. | Zbl 1013.65103
[22] , and, 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. | Zbl 0957.78023
[23] , Reversible adaptive regularization: perturbed Kepler motion and classical atomic trajectories. Philos. Trans. Roy. Soc. London Ser. A 357 (1999) 1101-1134. | Zbl 0933.65144
[24] and, Symplectic discretization for Maxwell's equations. J. Math. Computing 25 (2001) 1-21.
[25] , 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.
[26] ,, and, An efficient local time stepping-discontinuous Galerkin scheme for adaptive transient computations. Technical report 2001-13, Rensselaer Polytechnic Institute (2001).
[27] , A new finite volume scheme for solving Maxwell's system. COMPEL 19 (2000) 913-931. | Zbl 0994.78021
[28] , and, High-order symplectic integration methods for finite element solutions to time dependent Maxwell equations. IEEE Trans. Antennas Propagation 52 (2004) 2190-2195.
[29] and, Numerical Hamiltonian Problems, Chapman and Hall, London, UK (1994). | MR 1270017 | Zbl 0816.65042
[30] and, A comparative study of characteristic-based algorithms for the Maxwell equations. J. Comput. Phys. 125 (1996) 378-394. | Zbl 0848.65087
[31] , Application of the discontinuous Galerkin method to Maxwell’s equations using unstructured polymorphic -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. | Zbl 0957.78011
[32] , 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.
[33] , Numerical solution of initial boundary value problems involving Maxwell's equations in isotropic media. IEEE Trans. Antennas Propagation 16 (1966) 302-307.