Abstract
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 the cell interface through a simple hybrid gas distribution function. Due to the intrinsic connection between the gaskinetic 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.
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Arnold, D. N., Brezzi, F., Cockbum, B. and Marini, L. D., 2002, “Unified Analysis of Discontinuous Galerkin Methods for Elliptic Problems,”SIAM J. Numer. Anal. Vol. 39, p. 1749.
Bassi, F. and Rebay, S., 1997, “A High-order Accurate Discontinuous Finite Element Method for the Numerical Solution of the Compressible Navier-Stokes Equations,”J. Comput. Phys. pp. 131–267.
Baumann, C. E. and Oden, J. T., 1999, “A Discontinuoushp Finite Element Method for the Euler and Navier-Stokes Equations,”Int. J. Numer. Methods in Fluids. Vol. 31, p. 79.
Bhatnagar, P. L., Gross, E. P. and Krook, M., 1954, “A Model for Collision Processes in gases I: Small Amplitude Rocesses in Charged and Neutral Onecomponent Systems,”Phys. Rev. Vol. 94, p. 511.
Chou, S. Y. and Baganoff, D., 1997, “Kinetic Flux-Vector Splitting for the Navier-Stokes Equations,”Comput. Phys. Vol. 130, p. 217.
Cockbum, B., Karniadakis, G. E. and Shu, C. W., 2000, “The Development of Discontinuous Galerkin Methods,” Discontinuous Galerkin Methods: Theory, Computation and Applications, Springer, Berlin.
Cockbum, B. and Shu, C. W., 1989a, “TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Scalar Conservation Laws II: General framework,”Math. Comp. Vol 52, p. 411.
Cockbum, B., Lin, S. Y. and Shu, C. W., 1989b, “TVB Runge-Kurta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws III: One Dimensional Systems,”J. Comput. Phys. Vol. 84, p. 90.
Cockbum, B., Hou S. and Shu, C. W., 1990, “TVB Runge-Kutta Local Projection Discontinuous Galerkin Finite Element Method for Conservation Laws IV: The Multidimensional Case,”Math. Comp. Vol. 54, p. 545.
Cockburn, B. and Shu, C. W., 1998, “The Runge-Kutta Discontinuous Galerkin Method for Conservation Laws V: Multidimensional Systems,”J. Compta. Phys. Vol. 141, p. 199.
Cockburn, B. and Shu, C. W., 1998, “The Local Discontinuous Galerkin Method for Time-Dependent Convection-diffusion Systems,”SIAM J. Numer. Anal. Vol. 35, p. 2440
Cockburn, B. and Shu, C. W. 2001, “Runge-Kutta Discon-tinuous Galerkin Method for Convection-Dominated Problems,”J. Sci. Comput. Vol. 16, p. 173.
Hakkinen, R. J., and Greber, L., Trilling, L. and Abarbanel, S. S., 1959, “The Interaction of an Oblique Shock Wave with a Laminar Boundary Layer,”NASA Memo. 2-18-59W, NASA.
Ohwada, T., 2002, “On the Construction of Kinetic Schemes,”J. Comput. Phys. Vol. 177, p. 156.
Ohwada, T. and Fukata, S., “Simple Derivation of High-resolution Schemes for Compressible Flows by Kinetic Approach,”J. Comput. Phys. Vol. 211, p. 424.
Qiu, J. and Shu, C. W., 2004, “Hermite WENO Schemes and Their Application as Limiters for Runge-Kutta Discontinuous Galerkin Method: One Dimensional Case,”J. Comput. Phys. Vol. 193, p. 115.
Shu, C. W. and Osher, S., 1989, “Efficient Implementation of Essential Non-oscillatory Shock Capturing-Schemes,”II. J. Comput. Phys. Vol. 83, p. 32.
Tang, H. Z. and Warnecke, G., 2005, “A Runge-Kutta Discontinuous Galerkin Method for the Euler Equations,”Computers & Fluids, Vol. 34, p. 375.
van Leer, B., 1977, “Towards the Ultimate Conservative Difference Scheme IV. A New Approach to Numerical Convection,”J. Comput. Phys. Vol. 23, p. 276.
van Leer, B. and Nomura, S., 2005 “Discontinuous Galerkin for Diffusion,”AIAA-2005-5108, 17th AIAA Computational Fluid Dynamics Conference.
Xu, K., 1998, “Gas-kinetic Schemes for Unsteady Compressible flow Simulations,”VKI for Fluid Dynamics Lecture Series 1998-03.
Xu, K., 2001, “A Gas-kinetic BGK Scheme for the Navier-Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method,”J. Comput. Phys. Vol. 171, p. 289.
Xu, K. and Jameson, A., 1995, “Gas-Kinetic Relaxation (BGK-type) Schemes for the Compressible Euler Equations,”AIAA-95-1736, 12th AIAA Computational Fluid Dynamics Conference.
Xu, K. and Li, Z. W., 2001, “Dissipative Mechanism in Godunov-type Schemes,”Int. J. Numer. Methods in Fluids, Vol. 37, p. 1.
Xu, K., Mao, M. L. and Tang, L., “A Multidimensional Gas-kinetic BGK Scheme for Hypersonic Viscous flow,”J. Comput. Phys. Vol. 203, p. 405.
Xu, K., 2004, “Discontinuous Galerkin BGK Method for Viscous flow Equations: One-Dimensional Systems,”SIAMJ. Sci. Comput. Vol. 23, p. 1941.
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Liu, H., Xu, K. A gas-kinetic discontinuous Galerkin method for viscous flow equations. J Mech Sci Technol 21, 1344 (2007). https://doi.org/10.1007/BF03177419
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DOI: https://doi.org/10.1007/BF03177419