Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-10T14:38:00.871Z Has data issue: false hasContentIssue false
  • English
  • Français

A High-Order Accurate Gas-Kinetic Scheme for One- and Two-Dimensional Flow Simulation

Published online by Cambridge University Press:  03 June 2015

Na Liu  and
Huazhong Tang
Na Liu*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
Huazhong Tang*
Affiliation:
HEDPS, CAPT & LMAM, School of Mathematical Sciences, Peking University, Beijing 100871, P.R. China
*
Email:nliu@pku.edu.cn
Corresponding author.Email:hztang@math.pku.edu.cn
Get access

Abstract

This paper develops a high-order accurate gas-kinetic scheme in the framework of the finite volume method for the one- and two-dimensional flow simulations, which is an extension of the third-order accurate gas-kinetic scheme [Q.B. Li, K. Xu, and S. Fu, J. Comput. Phys., 229(2010), 6715-6731] and the second-order accurate gas-kinetic scheme [K. Xu, J. Comput. Phys., 171(2001), 289-335]. It is formed by two parts: quartic polynomial reconstruction of the macroscopic variables and fourth-order accurate flux evolution. The first part reconstructs a piecewise cell-center based quartic polynomial and a cell-vertex based quartic polynomial according to the “initial” cell average approximation of macroscopic variables to recover locally the non-equilibrium and equilibrium single particle velocity distribution functions around the cell interface. It is in view of the fact that all macroscopic variables become moments of a single particle velocity distribution function in the gas-kinetic theory. The generalized moment limiter is employed there to suppress the possible numerical oscillation. In the second part, the macroscopic flux at the cell interface is evolved in fourth-order accuracy by means of the simple particle transport mechanism in the microscopic level, i.e. free transport and the Bhatnagar-Gross-Krook (BGK) collisions. In other words, the fourth-order flux evolution is based on the solution (i.e. the particle velocity distribution function) of the BGK model for the Boltzmann equation. Several 1D and 2D test problems are numerically solved by using the proposed high-order accurate gas-kinetic scheme. By comparing with the exact solutions or the numerical solutions obtained the second-order or third-order accurate gas-kinetic scheme, the computations demonstrate that our scheme is effective and accurate for simulating invisid and viscous fluid flows, and the accuracy of the high-order GKS depends on the choice of the (numerical) collision time.

Keywords

76M1276M2576N15Gas-kinetic schemeinitial reconstructionBGK modelBoltzmann equationvelocity distribution function
Type
Research Article
Information
Communications in Computational Physics , Volume 15 , Issue 4 , April 2014 , pp. 911 - 943
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Biswas, R., Devine, K.D., and Flaherty, J.E.Parallel, adaptive finite element methods for conservation laws. Appl. Numer. Math., 14:255–283, 1994.Google Scholar
[2]Cockburn, B., Karniadakis, G.E., and Shu, C.-W.Discontinuous Galerkin Methods: Theory, Computation and Applications. Springer, 2000.Google Scholar
[3]Godunov, S.K.A finite-difference method for the numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat. Sb., 47:271–306, 1959.Google Scholar
[4]Hakkinen, R. J., Greber, I., Trilling, L., and Abarbanel, S. S.The interaction of an oblique shock wave with a laminar boundary layer. NASA, Memo 2-18-59W, 1959.Google Scholar
[5]EE Han, , Li, J.Q., and Tang, H.Z.Accuracy of the adaptive GRP scheme and the simulation of 2-D Riemann problems for compressible Euler equations. Commun. Comput. Phys., 10:577– 606, 2011.Google Scholar
[6]Harten, A.High resolution schemes for hyperbolic conservation laws. J. Comput. Phys., 49:357–393, 1983.Google Scholar
[7]Harten, A., Engquist, B., Osher, S., and Chakravarthy, S.R.Uniformly high order accurate essentially non-oscillatory schemes. J. Comput. Phys., 71:231–303, 1987.Google Scholar
[8]He, P. and Tang, H.Z.An adaptive moving mesh method for two-dimensional relativistic hydrodynamics. Commun. Comput. Phys., 11:114–146, 2012.Google Scholar
[9]Holden, H., Lie, K.A., and Risebro, N.H.An unconditionally stable method for the Euler equations. J. Comput. Phys., 150:76–96, 1999.Google Scholar
[10]Kogan, M.N.Rarefied Gas Dynamics. Plenum Press, New York, 1969.Google Scholar
[11]Krivodonova, L.Limiters for high-order discontinuous Galerkin methods. J. Comput. Phys., 226:879–896, 2007.Google Scholar
[12]Langseth, J.O. and LeVeque, R.J.A wave propagation method for three-dimensional hyperbolic conservation laws. J. Comput. Phys., 165:126–166, 2000.Google Scholar
[13]Lax, P.D. and Liu, X.D.Solution of two-dimensional Riemann problem of gas dynamics by positive schemes. SIAM J. Sci. Comput., 19:319–340, 1998.Google Scholar
[14]LeVeque, R.J.Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, 2009.Google Scholar
[15]Li, Q.B., Xu, K., and Fu, S.A high-order gas-kinetic Navier-Stokes flow solver. J. Comput. Phys., 229(19):6715–6731, 2010.Google Scholar
[16]Liu, X.D., Osher, S., and Chan, T.Weighted essentially non-oscillatory schemes. J. Comput. Phys., 115:200–212, 1994.Google Scholar
[17]Liu, Y., Vinokur, M., and Wang, Z.J.Spectral difference method for unstructured grids I: Basic formulation. J. Comput. Phys., 216:780–801, 2006.Google Scholar
[18]Mandal, J.C. and Despande, S.M.Kinetic flux vector splitting for the Euler equations. Comput. & Fluids, 23:447–478, 1994.CrossRefGoogle Scholar
[19]Shu, C.-W.High order weighted essentially nonoscillatory schemes for convection dominated problems. SIAM Rev., 51(1):82–126, 2009.Google Scholar
[20]Shu, C.-W. and Osher, S.Efficient implementation of essentially non-oscillatory shock-capturing schemes, II. J. Comput. Phys., 83:32–78, 1989.Google Scholar
[21]Tang, H.Z. and Tang, T.Adaptive mesh methods for one- and two-dimensional hyperbolic conservation laws. SIAM J. Numer. Anal., 41:487–515, 2003.Google Scholar
[22]Tang, H.Z. and Wu, H.M.High resolution KFVS finite volume methods and its applications in CFDs. Chinese J. Numer. Math. Appl., 21:93–103, 1999.Google Scholar
[23]Tang, H.Z. and Wu, H.M.Kinetic flux vector splitting for radiation hydrodynamical equations. Comput. & Fluids, 29:917–934, 2000.Google Scholar
[24]Tang, H.Z. and Xu, K.A high-order gas-kinetic method for multidimensional ideal magne-tohydrodynamics. J. Comput. Phys., 165:69–88, 2000.Google Scholar
[25]Tang, H.Z. and Xu, K.On positivity of a class of flux-vector splitting methods I. Explicit difference schemes. Math. Num. Sin., 23:469–476, 2001.Google Scholar
[26]Tang, T. and Xu, K.Gas-kinetic schemes for the compressible Euler equations: Positivity-preserving analysis. Z. Angew. Math. Phys., 50:258–281, 1999.Google Scholar
[27]Toro, E.F.Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 3rd edition, 2009.Google Scholar
[28]van Leer, B.Towards the ultimate conservative difference scheme V: a second order sequel to Godunov’s method. J. Comput. Phys., 32:101–136, 1979.Google Scholar
[29]van Leer, B.Upwind and high-resolution methods for compressible flow: From donor cell to residual-distribution schemes. Commun. Comput. Phys., 1:192–206, 2006.Google Scholar
[30]Wang, Z.J.Spectral (finite) volume method for conservation laws on unstructured grids: basic formulation. J. Comput. Phys., 178:210–251, 2002.Google Scholar
[31]Woodward, P. and Colella, P.Numerical simulations of two-dimensional fluid flow with strong shocks. J. Comput. Phys., 54:115–173, 1984.CrossRefGoogle Scholar
[32]Woodward, P. and Colella, P.The piecewise parabolic method (PPM) for gas-dynamical simulations. J. Comput. Phys., 54:174–201, 1984.Google Scholar
[33]Xu, K. Gas-kinetic schemes for unsteady compressible flow simulations. Technical Report 1998-03, VKI for Fluid Dynamics Lecture Series, 1998.Google Scholar
[34]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.Google Scholar
[35]Yang, M. and Wang, Z.J.A parameter-free generalized moment limiter for high-order methods on unstructured grids. Adv. Appl. Math. Mech., 1:451–480, 2009.Google Scholar