A gas-kinetic theory based multidimensional high-order method for the compressible Navier-Stokes solutions 
期刊名称Acta Mechanica Sinica
作者Xiaodong Ren; Kun Xu; Wei Shyy
摘要This paper presents a gas-kinetic theory based multidimensional high-order method for the compressible Naiver-Stokes solutions. In our previous study, a spatially and temporally dependent third-order flux scheme with the use of a third-order gas distribution function is employed. However, the third-order flux scheme is quite complicated and less robust than the second-order scheme. In order to reduce its complexity and improve its robustness, the second-order flux scheme is adopted instead in this paper, while the temporal order of method is maintained by using a two stage temporal discretization. In addition, its CPU cost is relatively lower than the previous scheme. Several test cases in two and three dimensions, containing high Mach number compressible flows and low speed high Reynolds number laminar flows, are presented to demonstrate the method capacity.
关键词Discontinuous Galerkin; Two-stage temporal discretization; Gas-kinetic theory
参考文献1. Cockburn, B., Shu, C. -W.:The Runge-Kutta discontinuous Galerkin method for conservation laws V. J. Comput. Phys. 141, 199-224 (1998)  
2. Reed, W. H., Hill, T.:Triangular mesh methods for the neutron transport equation. Los Alamos Report LA-UR-73-479 (1973)
3. Lasaint, P., Raviart, P. A.:On a finite element method for solving the neutron transport equation. Math. Asp. Finite Elem. Partial Differ. Equ. S4, 89-123 (1974)
4. Chavent, G., Salzano, G.:A finite-element method for the 1-D water flooding problem with gravity. J. Comput. Phys. 45, 307-344 (1982)  
5. Cockburn, B., Shu, C. -W.:TVB Runge-Kutta local projection discontinuous Galerkin finite element method for conservation laws b. General framework. Math. Comput. 52, 411-435 (1989)
6. Xu, Y., Shu, C. -W.:Local discontinuous Galerkin methods for high-order time-dependent partial differential equations. Commun. Comput. Phys. 7, 1-46 (2010)
7. Bassi, F., Rebay, S.:A high-order accurate discontinuous finite element method for the numerical solution of the compressible Navier-Stokes equations. J. Comput. Phys. 131, 267-279 (1997)  
8. Cockburn, B., Shu, C. W.:The local discontinuous Galerkin method for time-dependent convection-diffusion systems. Siam J. Numer. Anal. 35, 2440-2463 (1998)  
9. Kim, S. S., Kim, C., Rho, O. H., et al.:Cures for the shock instability:development of a shock-stable Roe scheme. J. Comput. Phys. 185, 342-374 (2003)  
10. Ren, X. D., Gu, C. W., Li, X. S.:Role of the momentum interpolation mechanism of the Roe scheme in shock instability. https://arxiv.org/vc/arxiv/papers/1509/1509.02776v1.pdf (2016)
11. Li, X. -S., Gu, C. -W.:Mechanism of Roe-type schemes for all-speed flows and its application. Comput. Fluids 86, 56-70 (2013)  
12. Li, X. -S.:Uniform algorithm for all-speed shock-capturing schemes. Int. J. Comput. Fluid Dyn. 28, 329-338 (2014)  
13. Xu, K.:A gas-kinetic BGK scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method. J. Comput. Phys. 171, 289-335 (2001)  
14. Xu, K.:Discontinuous Galerkin BGK method for viscous flow equations:One-dimensional systems. Siam J. Sci. Comput. 25, 1941-1963 (2004)  
15. Liu, H., Xu, K.:A Runge-Kutta discontinuous Galerkin method for viscous flow equations. J. Comput. Phys. 224, 1223-1242 (2007)  
16. Ni, G., Jiang, S., Xu, K.:A DGBGK scheme based on WENO limiters for viscous and inviscid flows. J. Comput. Phys. 227, 5799-5815 (2008)  
17. Luo, H., Luo, L. Q., Xu, K.:A discontinuous Galerkin method based on a BGK scheme for the Navier-Stokes equations on arbitrary grids. Adv. Appl. Math. Mech. 1, 301-318 (2009)
18. Ren, X., Xu, K., Shyy, W., et al.:A multi-dimensional high-order discontinuous Galerkin method based on gas kinetic theory for viscous flow computations. J. Comput. Phys. 292, 176-193 (2015)  
19. Pan, L., Xu, K., Li, Q., et al.:An efficient and accurate two-stage fourth-order gas-kinetic scheme for the Euler and Navier-Stokes equations. J. Comput. Phys. 326, 197-221 (2016)  
20. Luo, H., Baum, J. D., Löhner, R.:A discontinuous Galerkin method based on a Taylor basis for the compressible flows on arbitrary grids. J. Comput. Phys. 227, 8875-8893 (2008)  
21. Luo, J., Xu, K.:A high-order multidimensional gas-kinetic scheme for hydrodynamic equations. Sci. China Technol. Sci. 56, 2370-2384 (2013)  
22. Guo, Z., Xu, K., Wang, R.:Discrete unified gas kinetic scheme for all Knudsen number flows:low-speed isothermal case. Phys. Rev. E 88, 033305 (2013)  
23. Wieting, A. R.:Experimental study of shock wave interface heating on a cylindrical leading edge. NASA-TM-100484 (1987)
24. Daru, V., Tenaud, C.:High order one-step monotonicity-preserving schemes for unsteady compressible flow calculations. J. Comput. Phys. 193, 563-594 (2004)  
25. Kim, K. H., Kim, C.:Accurate, efficient and monotonic numerical methods for multi-dimensional compressible flows:part b:multi-dimensional limiting process. J. Comput. Phys. 208, 570-615 (2005)  
26. Albensoeder, S., Kuhlmann, H. C.:Accurate three-dimensional liddriven cavity flow. J. Comput. Phys 206, 536-558 (2005)  
27. Wong, K. L., Baker, A. J.:A 3D incompressible Navier-Stokes velocity-vorticity weak form finite element algorithm. Int. J. Numer. Methods Fluids 38, 99-123 (2002)  
基金项目The current work is supported by HKUST research fund PROVOST 13SC01.
作者地址1 Key Laboratory for Thermal Science and Power Engineering of Ministry of Education, Department of Thermal Engineering, Tsinghua University, Beijing 100084, China;
2 Department of Mathematics, School of Science, The Hong Kong University of Science and Technology, Hong Kong, China;
3 Department of Mechanical and Aerospace Engineering, School of Engineering, The Hong Kong University of Science and Technology, Hong Kong, China

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