A Runge-Kutta discontinuous Galerkin method for viscous flow equations

This paper presents a Runge-Kutta discontinuous Galerkin (RKDG) method for viscous flow computation. The construction of the RKDG method is based on a gas-kinetic formulation, which not only couples the convective and dissipative terms together, but also includes both discontinuous and continuous representation in the flux evaluation at a cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gas-kinetic BGK model and the Navier-Stokes equations, the Navier-Stokes flux is automatically obtained by the present method. Numerical examples for both one dimensional (1D) and two dimensional (2D) compressible viscous flows are presented to demonstrate the accuracy and shock capturing capability of the current RKDG method.

[1]  George Em Karniadakis,et al.  The Development of Discontinuous Galerkin Methods , 2000 .

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

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

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

[5]  I. Flügge-Lotz,et al.  The interaction of a shock wave with a laminar boundary layer , 1968 .

[6]  P. Bhatnagar,et al.  A Model for Collision Processes in Gases. I. Small Amplitude Processes in Charged and Neutral One-Component Systems , 1954 .

[7]  Leon Trilling,et al.  The Interaction of an Oblique Shock Wave with a Laminar Boundary Layer , 1959 .

[8]  Kun Xu,et al.  A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method , 2001 .

[9]  Taku Ohwada,et al.  On the Construction of Kinetic Schemes , 2002 .

[10]  Chi-Wang Shu.  TVB uniformly high-order schemes for conservation laws , 1987 .

[11]  Gerald Warnecke,et al.  A Runge–Kutta discontinuous Galerkin method for the Euler equations , 2005 .

[12]  Kun Xu,et al.  Dissipative mechanism in Godunov‐type schemes , 2001 .

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

[14]  Chi-Wang Shu,et al.  Runge–Kutta Discontinuous Galerkin Methods for Convection-Dominated Problems , 2001, J. Sci. Comput..

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

[16]  Antony Jameson,et al.  Gas-kinetic relaxation (BGK-type) schemes for the compressible Euler equations , 1995 .

[17]  Bram Van Leer,et al.  Discontinuous Galerkin for Diffusion , 2005 .

[18]  Kun Xu,et al.  Discontinuous Galerkin BGK Method for Viscous Flow Equations: One-Dimensional Systems , 2004, SIAM J. Sci. Comput..

[19]  Kun Xu,et al.  Gas-kinetic schemes for unsteady compressible flow simulations , 1998 .

[20]  D. Baganoff,et al.  Kinetic Flux-Vector Splitting for the Navier-Stokes Equations , 1997 .

[21]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme. IV. A new approach to numerical convection , 1977 .

[22]  Kun Xu,et al.  A multidimensional gas-kinetic BGK scheme for hypersonic viscous flow , 2005 .

[23]  Chi-Wang Shu,et al.  The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems , 1998 .

[24]  Taku Ohwada,et al.  Simple derivation of high-resolution schemes for compressible flows by kinetic approach , 2006 .

[25]  Iizln.  Gas-Kinetic Relaxation ( BGK-Type ) Schemes for the Compressible Euler Equations , 2004 .

[26]  Douglas N. Arnold,et al.  Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems , 2001, SIAM J. Numer. Anal..

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

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

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