Journal of Scientific Computing

, Volume 22, Issue 1–3, pp 443–477 | Cite as

A Discontinuous Galerkin Method for Linear Symmetric Hyperbolic Systems in Inhomogeneous Media

  • Peter Monk
  • Gerard R. Richter


The Discontinuous Galerkin (DG) method provides a powerful tool for approximating hyperbolic problems. Here we derive a new space-time DG method for linear time dependent hyperbolic problems written as a symmetric system (including the wave equation and Maxwell’s equations). The main features of the scheme are that it can handle inhomogeneous media, and can be time-stepped by solving a sequence of small linear systems resulting from applying the method on small collections of space-time elements. We show that the method is stable provided the space-time grid is appropriately constructed (this corresponds to the usual time-step restriction for explicit methods, but applied locally) and give an error analysis of the scheme. We also provide some simple numerical tests of the algorithm applied to the wave equation in two space dimensions (plus time).


Discontinuous Galerkin finite elements hyperbolic explicit 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Cockburn, B., Shu, C. 1989TVD Runge-Kutta projection discontinuous Galerkin finite element method for conservation laws II: General frameworkMath.Comput.52411435Google Scholar
  2. Driscoll, T., Fornberg, B. 1999Block pseudospectral methods for Maxwell’s equations II: Two-dimensional discontinuous-coefficient caseSIAM J.Numer.Anal.2111461167Google Scholar
  3. Engquist, B., Majda, A. 1977Absorbing boundary conditions for the numerical simulation of wavesMath.Comput.31629651Google Scholar
  4. Erickson, J., Guoy, D., Sullivan, J., and Üngör, A. (2002). Building space–time meshes over arbitrary spatial domains. In Proc. 11th Int. Meshing Roundtable, Sandia, pp. 391–402.Google Scholar
  5. Falk, R., Richter, G. 1999Explicit finite element methods for symmetric hyperbolic equationsSIAM J.Numer.Anal.36935952CrossRefGoogle Scholar
  6. Friedrichs, K. 1958Symmetric positive linear differential equationsComm. Pure Appl. Math.11333418Google Scholar
  7. Hesthaven, J., Warburton, T. 2002Nodal high-order methods on unstructured grids-I.Time-domain solution of Maxwell’s equationsJ.Comput.Phys.181186221CrossRefGoogle Scholar
  8. Hu, F., Hussaini, M., Rasetarinera, P. 1999An analysis of the discontinuous Galerkin method for wave propagation problemsJ.Comput.Phys.151921946CrossRefGoogle Scholar
  9. Hughes, T., Hulbert, G. 1988Space–time finite element methods for elastodynamics: Formulations and error estimatesComput.Meth.Appl.Mech.Eng.66339363CrossRefGoogle Scholar
  10. Johnson, C., Pitkäranta, J. 1986An analysis of the discontinuous Galerkin method for a scalar hyperbolic equationMath.Comput.47285312Google Scholar
  11. Lesaint, P., Raviart, P. 1974On a finite element method for solving the neutron transport equationdeBoor, C. eds. Mathematical Aspects of Finite Element Methods in Partial Differential EquationsAcademic PressNew York89123Google Scholar
  12. Monk, P. 1992A comparison of three mixed methods for the time dependent Maxwell equationsSIAM J. Sci. Stat. Comput.1310971122CrossRefGoogle Scholar
  13. Reed, W., Hill, T. 1973Triangular Mesh Methods for the Neutron Transport Equation. Technical Report LA-UR-73-479Los Alamos National LaboratoryLos Alamos, New Mexico, USAGoogle Scholar
  14. Richter, G. 1988An optimal order estimate for the discontinuous Galerkin methodMath.Comput.507588Google Scholar
  15. Üngör, A., Sheffer, A. 2002Pitching tents in space–time: Mesh generation for discontinuous Galerkin method. Int.J.Found.Comput.Sci.13201221CrossRefGoogle Scholar
  16. Vila, J.-P., Villedieu, P. 2003Convergence of an explicit finite volume scheme for first order symmetric systemsNumer.Math.94573602CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Computer Science DepartmentRutgers UniversityPiscatawayUSA

Personalised recommendations