DGTD methods using modal basis functions and symplectic local time-stepping
Application to wave propagation problems
Keywords:
waves, acoustics, Maxwell’s system, discontinuous Galerkin methods, mass matrix condition number, symplectic schemes, energy conservation, local time-steppingAbstract
The discontinuous Galerkin time domain (DGTD) methods are now widely used for the solution of wave propagation problems. Able to deal with unstructured meshes past complex geometries, they remain fully explicit with easy parallelization and extension to high orders of accuracy. Still, modal or nodal local basis functions have to be chosen carefully to obtain actual numerical accuracy. Concerning time discretization, explicit non-dissipative energypreserving time-schemes exist, but their stability limit remains linked to the smallest element size in the mesh. Symplectic algorithms, based on local-time stepping or local implicit scheme formulations, can lead to dramatic reductions of computational time, which is shown here on two-dimensional acoustics problems.
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References
Bécache E., Joly P., Rodríguez J., “ Space-time mesh refinement for elastodynamics. Numerical
results.”, Comput. Methods Appl. Mech. Eng., vol. 194, n 2-5, p. 355-366, 2005.
Canouet N., Fezoui L., Piperno S., “ A discontinuous Galerkin method for 3D Maxwell’s equation
on non-conforming grids”, in G. C. Cohen, E. Heikkola, P. Joly, P. Neittaanmäki (eds),
WAVES 2003, The Sixth International Conference on Mathematical and Numerical Aspects
of Wave Propagation, Springer-Verlag, Jyväskylä, Finland, p. 389-394, 2003.
Cioni J.-P., Fezoui L., Anne L., Poupaud F., “ A parallel FVTD Maxwell solver using 3D unstructured
meshes”, 13th annual review of progress in applied computational electromagnetics,
Monterey, California, 1997.
Cockburn B., Karniadakis G. E., Shu C.-W. (eds), Discontinuous Galerkin methods. Theory,
computation and applications., vol. 11 of Lecture Notes in Computational Science and
Engineering, Springer-Verlag, Berlin, 2000.
Elmkies A., 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., vol. t. 325, n 11,
p. 1217-1222, 1997.
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”, RAIRO Modél. Math. Anal. Numér., n.d. to appear.
Fouquet T., Raffinement de maillage spatio-temporel pour les équations de Maxwell, PhD thesis,
University Paris 9, France, 2000.
Hardy D. J., Okunbor D. I., Skeel R. D., “ Symplectic variable step size integration for N-body
problems”, Appl. Numer. Math., vol. 29, p. 19-30, 1999.
Hesthaven J., Teng C., “ Stable spectral methods on tetrahedral elements”, J. Sci. Comput., vol.
, p. 2352-2380, 2000.
Hesthaven J., Warburton T., “ Nodal high-order methods on unstructured grids. I: Time-domain
solution of Maxwell’s equations”, J. Comput. Phys., vol. 181, p. 186-221, 2002a.
Hesthaven J., Warburton T., “ Nodal high-order methods on unstructured grids. I: Time-domain
solution of Maxwell’s equations”, J. Comput. Phys., vol. 181, n 1, p. 186-221, 2002b.
Hirono T., Lui W. W., Yokoyama K., “ Time-domain simulation of electromagnetic field using
a symplectic integrator”, IEEE Microwave Guided Wave Lett., vol. 7, p. 279-281, 1997.
Hirono T., LuiW.W., Yokoyama K., Seki S., “ Stability and numerical dispersion of symplectic
fourth-order time-domain schemes for optical field simulation”, J. Lightwave Tech., vol. 16,
p. 1915-1920, 1998.
Holder T., Leimkuhler B., Reich S., “ Explicit variable step-size and time-reversible integration”,
Appl. Numer. Math., vol. 39, p. 367-377, 2001.
Huang W., Leimkuhler B., “ The adaptive Verlet method”, J. Sci. Comput., vol. 18, p. 239-256,
Hyman J. M., Shashkov M., “ Mimetic discretizations for Maxwell’s equations”, J. Comput.
Phys., vol. 151, p. 881-909, 1999.
Joly P., Poirier C., “ A new second order 3D edge element on tetrahedra for time dependent
Maxwell’s equations”, in A. Bermudez, D. Gomez, C. Hazard, P. Joly, J.-E. Roberts (eds),
Fifth International Conference on Mathematical and Numerical Aspects of Wave Propagation,
SIAM, Santiago de Compostella, Spain, p. 842-847, July 10-14, 2000.
Kopriva D. A., Woodruff S. L., Hussaini M. Y., Discontinuous spectral element approximation
of Maxwell’s equations, vol. 11 of Lecture Notes in Computational Science and Engineering,
Springer-Verlag, Berlin, p. 355-362, 2000.
Lu X., Schmide R., “ Symplectic Discretization for Maxwell’s Equations”, J. of Math. and
Computing, 2001.
Piperno S., Schémas en éléments finis discontinus localement raffinés en espace et en temps
pour les équations de Maxwell 1D, Technical Report n 4986, INRIA, 2003.
Piperno S., Symplectic local time-stepping in non-dissipative DGTD methods applied to wave
propagation problems, Technical Report n 5643, INRIA, 2005.
Piperno S., “ Symplectic local time-stepping in non-dissipative DGTD methods applied to wave
propagation problems”, RAIRO Modél. Math. Anal. Numér., n.d. submitted for publicqtion.
Remaki M., “ A New Finite Volume Scheme for Solving Maxwell’s System”, COMPEL, vol.
, n 3, p. 913-931, 2000.
Rieben R., White D., Rodrigue G., “ High-Order Symplectic Integration Methods for Finite
Element Solutions to Time Dependent Maxwell Equations”, IEEE Trans. Antennas and
Propagation, vol. 52, p. 2190-2195, 2004.
Sanz-Serna J. M., Calvo M. P., Numerical Hamiltonian Problems, Chapman and Hall, London,
U.K., 1994.
Shang J., Fithen R., “ A Comparative Study of Characteristic-Based Algorithms for the Maxwell
equations”, J. Comput. Phys., vol. 125, p. 378-394, 1996.
Warburton T., Application of the discontinuous Galerkin method to Maxwell’s equations using
unstructured polymorphic hp-finite elements, vol. 11 of Lecture Notes in Computational
Science and Engineering, Springer-Verlag, Berlin, p. 451-458, 2000.
Warburton T., “ Spurious solutions and the Discontinuous Galerkin method on non-conforming
meshes”, WAVES 2005, The Seventh International Conference on Mathematical and Numerical
Aspects of Wave Propagation, Providence, RI., p. 405-407, 2005.
Yee K. S., “ Numerical solution of initial boundary value problems involving Maxwell’s equations
in isotropic media”, IEEE Trans. Antennas and Propagation, vol. AP-16, p. 302-307,
