Arbitrary high order discontinuous Galerkin schemes based on the GRP method for compressible Euler equations

Combined with the discontinuous Galerkin (DG) framework, the generalized Riemann problem (GRP) method is applied to design a GRP-DG scheme with high order accuracy for compressible Euler equations. Since numerical fluxes with second order accuracy in time are derived by the GRP method, the reconstruction steps for physical variables in the new scheme are halved compared with the traditional Runge-Kutta discontinuous Galerkin (RK-DG) scheme. The numerical results are also improved due to more introduced physical information. Several numerical examples verify the validity of the proposed schemes.

[1]  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.

[2]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[3]  Guido Kanschat,et al.  The local discontinuous Galerkin method for the Oseen equations , 2003, Math. Comput..

[4]  Jiequan Li,et al.  The generalized Riemann problems for compressible fluid flows: Towards high order , 2014, J. Comput. Phys..

[5]  Jiequan Li,et al.  An adaptive GRP scheme for compressible fluid flows , 2010, J. Comput. Phys..

[6]  P. Woodward,et al.  The numerical simulation of two-dimensional fluid flow with strong shocks , 1984 .

[7]  Jiequan Li,et al.  Hyperbolic balance laws: Riemann invariants and the generalized Riemann problem , 2007, Numerische Mathematik.

[8]  Chi-Wang Shu.  Total-variation-diminishing time discretizations , 1988 .

[9]  Wei Cai,et al.  Uniform high-order spectral methods for one- and two-dimensional Euler equations , 1991 .

[10]  J. Falcovitz,et al.  An upwind second-order scheme for compressible duct flows , 1986 .

[11]  W. H. Reed,et al.  Triangular mesh methods for the neutron transport equation , 1973 .

[12]  P. Lax,et al.  Systems of conservation laws , 1960 .

[13]  Michael Dumbser,et al.  Efficient, high accuracy ADER-WENO schemes for hydrodynamics and divergence-free magnetohydrodynamics , 2008, Journal of Computational Physics.

[14]  P. Roe,et al.  On Godunov-type methods near low densities , 1991 .

[15]  Tiegang Liu,et al.  Implementation of the GRP scheme for computing radially symmetric compressible fluid flows , 2009, J. Comput. Phys..

[16]  J. Tinsley Oden,et al.  A discontinuous hp finite element method for the Euler and Navier–Stokes equations , 1999 .

[17]  Michael Dumbser,et al.  Efficient implementation of ADER schemes for Euler and magnetohydrodynamical flows on structured meshes - Speed comparisons with Runge-Kutta methods , 2013, J. Comput. Phys..

[18]  S. Rebay,et al.  A High-Order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations , 1997 .

[19]  Jiequan Li,et al.  The generalized Riemann problem method for the shallow water equations with bottom topography , 2006 .

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

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

[22]  Joseph Falcovitz,et al.  Generalized Riemann Problems in Computational Fluid Dynamics , 2003 .

[23]  Michael Dumbser,et al.  A unified framework for the construction of one-step finite volume and discontinuous Galerkin schemes on unstructured meshes , 2008, J. Comput. Phys..

[24]  S. Rebay,et al.  High-Order Accurate Discontinuous Finite Element Solution of the 2D Euler Equations , 1997 .

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

[26]  Matania Ben-Artzi,et al.  The generalized Riemann problem for reactive flows , 1989 .

[27]  Gerald Warnecke,et al.  A direct Eulerian GRP scheme for compressible fluid flows , 2006, J. Comput. Phys..

[28]  Guido Kanschat,et al.  Local Discontinuous Galerkin Methods for the Stokes System , 2002, SIAM J. Numer. Anal..

[29]  Tiegang Liu,et al.  A note on the conservative schemes for the Euler equations , 2006, J. Comput. Phys..

[30]  George Em Karniadakis,et al.  A Discontinuous Galerkin Method for the Viscous MHD Equations , 1999 .

[31]  Zhongfeng Sun,et al.  Remark on the generalized Riemann problem method for compressible fluid flows , 2007, J. Comput. Phys..

[32]  E Rivoalen,et al.  Numerical Simulation of Axisymmetric Viscous Flows by Means of a Particle Method , 1999 .

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

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

[35]  J. Falcovitz,et al.  A second-order Godunov-type scheme for compressible fluid dynamics , 1984 .

[36]  Jiequan Li,et al.  Remapping-Free Adaptive GRP Method for Multi-Fluid Flows I: One Dimensional Euler Equations , 2014 .

[37]  George Em Karniadakis,et al.  Spectral/hp Methods for Viscous Compressible Flows on Unstructured 2D Meshes , 1998 .

[38]  Simulations of viscous supersonic flows on unstructured h-p meshes , 1997 .

[39]  George Em Karniadakis,et al.  A discontinuous Galerkin method for the Navier-Stokes equations , 1999 .

[40]  Eleuterio F. Toro,et al.  ADER schemes for three-dimensional non-linear hyperbolic systems , 2005 .

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

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

[43]  S. Osher,et al.  Efficient implementation of essentially non-oscillatory shock-capturing schemes,II , 1989 .

[44]  Chi-Wang Shu,et al.  Total variation diminishing Runge-Kutta schemes , 1998, Math. Comput..