• Zhang, Tie (Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries Northeastern University) ;
  • Liu, Jingna (Department of Mathematics and the State Key Laboratory of Synthetical Automation for Process Industries Northeastern University)
  • Received : 2013.05.17
  • Accepted : 2014.02.24
  • Published : 2014.07.01


We present a new space-time discontinuous Galerkin (DG) method for solving the time dependent, positive symmetric hyperbolic systems. The main feature of this DG method is that the discrete equations can be solved semi-explicitly, layer by layer, in time direction. For the partition made of triangle or rectangular meshes, we give the stability analysis of this DG method and derive the optimal error estimates in the DG-norm which is stronger than the $L_2$-norm. As application, the wave equation is considered and some numerical experiments are provided to illustrate the validity of this DG method.


discontinuous Galerkin method;first-order hyperbolic system;semi-explicit scheme;stability and error estimate


  1. A. Ern and J. L. Guermond, Discontinuous Galerkin methods for Friedrichs' systems, I. general theory, SIAM J. Numer. Anal. 44 (2006), no. 2, 753-778.
  2. B. Cockburn, G. E. Karniadakis, and C. W. Shu, Discontinuous Galerkin Methods, Theory, Computation and Applications, Lecture Notes Comput. Sci. Eng., Vol. 11, Springer-Verlag, Berlin, 2000.
  3. B. Cockburn, B. Dong, and J. Guzman, Optimal convergence of the original DG method for the transport-reaction equation on special meshes, SIAM J. Numer. Anal. 46 (2008), no. 3, 1250-1265.
  4. B. Cockburn and C. W. Shu, Runge-Kutta discontinuous Galerkin methods for convection-dominated problems (review article), J. Sci. Compu. 16 (2001), no. 3, 173-261.
  5. R. S. Falk and G. R. Richter, Explicit finite element methods for symmetric hyperbolic equations, SIAM J. Numer. Anal. 36 (1999), no. 3, 935-952.
  6. K. Friedrichs, Symmetric positive linear differential equations, Comm. Pure Appl. Math. 11 (1958), 333-418.
  7. C. Johnson, U. Navert, and J. Pitkaranta, Finite element methods for linear hyperbolic problems, Comput. Methods Appl. Mech. Engrg. 45 (1984), no. 1-3, 285-312.
  8. P. Monk and G. R. Richter, A discontinuous Galerkin method for linear symmetric hyperbolic systems in inhomogeneous media, J. Sci. Compu. 22 (2005), 443-477.
  9. T. E. Peterson, A note on the convergence of the discontinuous Galerkin method for a scalar hyperbolic equation, SIAM J. Numer. Anal. 28 (1991), no. 1, 133-140.
  10. G. Richter, An optimal-order error estimate for discontinuous Galerkin method, Math. Comp. 50 (1988), 75-88.
  11. R. Winther, Astable finite element method for first-order hyperbolic systems, Math. Comp. 36 (1981), 65-86.
  12. T. Zhang, Discontinuous Finite Element Theory and Method, Sincece Press, Beijing, 2012.
  13. D. N. Arnold, F. Brezzi, and B. Cockburn, et al., Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal. 39 (2001), no. 5, 1749-1779.
  14. P. G. Cairlet, The Finite Element Methods for Elliptic Problems, North-Holland Publish, Amsterdam, 1978.