Analysis and implementation of the gas-kinetic BGK scheme for computational gas dynamics

SUMMARY Gas-kinetic schemes based on the BGK model are proposed as an alternative evolution model which can cure some of the limitations of current Riemann solvers. To analyse the schemes, simple advection equations are reconstructed and solved using the gas-kinetic BGK model. Results for gas-dynamic application are also presented. The final flux function derived in this model is a combination of a gas-kinetic Lax‐Wendroff flux of viscous advection equations and kinetic flux vector splitting. These two basic schemes are coupled through a non-linear gas evolution process and it is found that this process always satisfies the entropy condition. Within the framework of the LED (local extremum diminishing) principle that local maxima should not increase and local minima should not decrease in interpolating physical quantities, several standard limiters are adopted to obtain initial interpolations so as to get higher-order BGK schemes. Comparisons for well-known test cases indicate that the gas-kinetic BGK scheme is a promising approach in the design of numerical schemes for hyperbolic conservation laws. # 1997 by John Wiley & Sons, Ltd.

[1]  A. Przekwas,et al.  A comparative study of advanced shock-capturing schemes applied to Burgers' equation , 1990 .

[2]  I. Bohachevsky,et al.  Finite difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics , 1959 .

[3]  C. Hirsch,et al.  Numerical Computation of Internal and External Flows. By C. HIRSCH. Wiley. Vol. 1, Fundamentals of Numerical Discretization. 1988. 515 pp. £60. Vol. 2, Computational Methods for Inviscid and Viscous Flows. 1990, 691 pp. £65. , 1991, Journal of Fluid Mechanics.

[4]  Philip L. Roe,et al.  Progress on multidimensional upwind Euler solvers for unstructured grids , 1991 .

[5]  J. C. Mandal,et al.  KINETIC FLUX VECTOR SPLITTING FOR EULER EQUATIONS , 1994 .

[6]  S. Zalesak.  Fully multidimensional flux-corrected transport algorithms for fluids , 1979 .

[7]  B. Grossman,et al.  A multi-dimensional kinetic-based upwind solver for the Euler equations , 1993 .

[8]  Antony Jameson,et al.  Positive schemes and shock modelling for compressible flows , 1995 .

[9]  B. V. Leer,et al.  Towards the Ultimate Conservative Difference Scheme , 1997 .

[10]  Elaine S. Oran,et al.  Numerical methods in reacting flows , 1987 .

[11]  P. Roe.  Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes , 1997 .

[12]  A. J. Przekwas,et al.  A comparative study of advanced shock-capturing shcemes applied to Burgers' equation , 1992 .

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

[14]  S. Osher,et al.  Uniformly High-Order Accurate Nonoscillatory Schemes. I , 1987 .

[15]  Bernd Einfeld.  On Godunov-type methods for gas dynamics , 1988 .

[16]  Antony Jameson,et al.  BGK-Based Schemes for the Simulation of Compressible Flow , 1996 .

[17]  Mikhail Naumovich Kogan,et al.  Rarefied Gas Dynamics , 1969 .

[18]  H. Gg. Wagner,et al.  Walter G. Vincenti und Charles H. Krüger, Jr.: Introduction to Physical Gas Dynamics. John Wiley & Sons, New York, London, Sidney 1965. 538 Seiten. Preis: 102/— , 1966, Berichte der Bunsengesellschaft für physikalische Chemie.

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

[20]  Jay P. Boris,et al.  Flux-corrected transport. I. SHASTA, a fluid transport algorithm that works , 1973 .

[21]  B. Perthame.  Second-order Boltzmann schemes for compressible Euler equations in one and two space dimensions , 1992 .

[22]  Antony Jameson,et al.  Gas-kinetic finite volume methods, flux-vector splitting, and artificial diffusion , 1995 .

[23]  P. Roe.  CHARACTERISTIC-BASED SCHEMES FOR THE EULER EQUATIONS , 1986 .

[24]  S. Osher,et al.  Uniformly high order accuracy essentially non-oscillatory schemes III , 1987 .

[25]  Philip L. Roe,et al.  A multidimensional flux function with applications to the Euler and Navier-Stokes equations , 1993 .

[26]  C. Angelopoulos.  High resolution schemes for hyperbolic conservation laws , 1992 .

[27]  P. Sweby.  High Resolution Schemes Using Flux Limiters for Hyperbolic Conservation Laws , 1984 .

[28]  Bram van Leer,et al.  Upwind-difference methods for aerodynamic problems governed by the Euler equations , 1985 .

[29]  P. Woodward,et al.  The Piecewise Parabolic Method (PPM) for Gas Dynamical Simulations , 1984 .

[30]  A. Jameson.  ANALYSIS AND DESIGN OF NUMERICAL SCHEMES FOR GAS DYNAMICS, 1: ARTIFICIAL DIFFUSION, UPWIND BIASING, LIMITERS AND THEIR EFFECT ON ACCURACY AND MULTIGRID CONVERGENCE , 1995 .

[31]  R. D. Richtmyer,et al.  Difference methods for initial-value problems , 1959 .

[32]  T. Teichmann,et al.  Introduction to physical gas dynamics , 1965 .

[33]  B. V. Leer,et al.  Towards the ultimate conservative difference scheme V. A second-order sequel to Godunov's method , 1979 .

[34]  Kun Xu,et al.  Numerical Navier-Stokes solutions from gas kinetic theory , 1994 .

[35]  Aaas News,et al.  Book Reviews , 1893, Buffalo Medical and Surgical Journal.

[36]  Kun Xu,et al.  Numerical hydrodynamics from gas-kinetic theory , 1993 .

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

[38]  R. LeVeque.  Numerical methods for conservation laws , 1990 .

[39]  P. Lax,et al.  On Upstream Differencing and Godunov-Type Schemes for Hyperbolic Conservation Laws , 1983 .