Arbitrary high order discontinuous Galerkin schemes

In this paper we apply the ADER one step time discretization to the Discontinuous Galerkin framework for hyperbolic conservation laws. In the case of linear hyperbolic systems we obtain a quadrature-free explicit single-step scheme of arbitrary order of accuracy in space and time on Cartesian and triangular meshes. The ADERDG scheme does not need more memory than a first order explicit Euler time-stepping scheme. This becomes possible because of an extensive use of the governing equations inside the numerical scheme. In the nonlinear case, quadrature of the ADER-DG scheme in space and time is performed with Gaussian quadrature formulae of suitable order of accuracy. We show numerical convergence results for the linearized Euler equations up to 10th order of accuracy in space and time on Cartesian and triangular meshes. Numerical results for the nonlinear Euler equations up to 6th order of accuracy in space and time are provided as well. In this paper we also show the possibility of applying a linear reconstruction operator of the order 3N +2 to the degrees of freedom of the DG method resulting in a numerical scheme of the order 3N + 3 on Cartesian grids where N is the order of the original basis functions before reconstruction.

[1]  R. Courant,et al.  On the solution of nonlinear hyperbolic differential equations by finite differences , 1952 .

[2]  A. Stroud.  Approximate calculation of multiple integrals , 1973 .

[3]  J. Butcher.  The numerical analysis of ordinary differential equations: Runge-Kutta and general linear methods , 1987 .

[4]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous galerkin finite element method for conservation laws. II: General framework , 1989 .

[5]  Chi-Wang Shu,et al.  TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws III: one-dimensional systems , 1989 .

[6]  Chi-Wang Shu,et al.  The Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws. IV. The multidimensional case , 1990 .

[7]  H. Atkins,et al.  Quadrature-Free Implementation of the Discontinuous Galerkin Method for Hyperbolic Equations , 1996 .

[8]  Chi-Wang Shu,et al.  The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V , 1998 .

[9]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[10]  Chaowei Hu,et al.  No . 98-32 Weighted Essentially Non-Oscillatory Schemes on Triangular Meshes , 1998 .

[11]  G. Karniadakis,et al.  Spectral/hp Element Methods for CFD , 1999 .

[12]  E. Toro,et al.  Solution of the generalized Riemann problem for advection–reaction equations , 2002, Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences.

[13]  Claus-Dieter Munz,et al.  ADER: A High-Order Approach for Linear Hyperbolic Systems in 2D , 2002, J. Sci. Comput..

[14]  Eleuterio F. Toro,et al.  ADER: Arbitrary High Order Godunov Approach , 2002, J. Sci. Comput..

[15]  M. Dumbser,et al.  On the Improvement of Efficiency and Storage Requirements of the Discontinuous Galerkin Method for Aeroacoustics , 2003 .

[16]  Endre Süli,et al.  Enhanced accuracy by post-processing for finite element methods for hyperbolic equations , 2003, Math. Comput..

[17]  Michael Dumbser,et al.  Fast high order ADER schemes for linear hyperbolic equations , 2004 .

[18]  Jianxian Qiu,et al.  Hermite WENO schemes and their application as limiters for Runge-Kutta discontinuous Galerkin method: one-dimensional case , 2004 .

[19]  Jennifer K. Ryan,et al.  Extension of a Post Processing Technique for the Discontinuous Galerkin Method for Hyperbolic Equations with Application to an Aeroacoustic Problem , 2005, SIAM J. Sci. Comput..

[20]  Michael Dumbser,et al.  ADER discontinuous Galerkin schemes for aeroacoustics , 2005 .

[21]  Michael Dumbser,et al.  The discontinuous Galerkin method with Lax-Wendroff type time discretizations , 2005 .