On the numerical solution of a BGK-type model for chemical reactions

In recent years, new suggestions and proposals concerning the mathematical modelling and applications of multi-component gaseous flows with chemical reactions have been published. A recent consistent BGK-type approach to reacting gas mixtures, according to Boltzmann-like kinetics for a bimolecular reversible chemical reaction has been introduced. In this paper we apply a numerical strategy based on time splitting techniques to simulate the reactive BGK equations. These techniques, widely used in the numerical approximation of the Boltzmann kinetic description, but not in the numerical analysis of the kinetic systems describing reacting gas mixtures, have the advantage of simplifying the problem by treating separately the convection step and collision step. Numerical tests of various splitting schemes on the classical BGK equation and several numerical results of the time-dependent Riemann problem for the reactive BGK system are presented.

[1]  P. Lions,et al.  Splitting Algorithms for the Sum of Two Nonlinear Operators , 1979 .

[2]  W. G. Bickley,et al.  Formulae for Numerical Differentiation , 1941, The Mathematical Gazette.

[3]  V. Giovangigli.  Multicomponent flow modeling , 1999 .

[4]  G. Sod.  A survey of several finite difference methods for systems of nonlinear hyperbolic conservation laws , 1978 .

[5]  Kazuo Aoki,et al.  Numerical analysis of gas flows condensing on its plane condensed phase on the basis of kinetic theory , 1990 .

[6]  P. Welander,et al.  ON THE TEMPERATURE JUMP IN A RAREFIED GAS , 1954 .

[7]  Francis Filbet,et al.  High order numerical methods for the space non-homogeneous Boltzmann equation , 2003 .

[8]  G. Spiga,et al.  Grad's distribution functions in the kinetic equations for a chemical reaction , 2002 .

[9]  C. Cercignani.  The Boltzmann equation and its applications , 1988 .

[10]  I. Prigogine,et al.  On the perturbation of Maxwell distribution function by chemical reactions in gases , 1949 .

[11]  T. Ohwada.  Higher Order Approximation Methods for the Boltzmann Equation , 1998 .

[12]  M. Pennacchio,et al.  An IMEX finite volume scheme for reactive Euler equations arising from kinetic theory , 2003 .

[13]  Maria Groppi,et al.  A Bhatnagar–Gross–Krook-type approach for chemically reacting gas mixtures , 2004 .

[14]  B. Perthame,et al.  A Consistent BGK-Type Model for Gas Mixtures , 2002 .

[15]  G. Ludwig,et al.  Boundary-Layer Theory with Dissociation and Ionization , 1960 .

[16]  G. Spiga,et al.  Kinetic approach to chemical reactions and inelastic transitions in a rarefied gas , 1999 .

[17]  Yoshio Sone,et al.  Kinetic Theory and Fluid Dynamics , 2002 .

[18]  C. Chu.  Kinetic‐Theoretic Description of the Formation of a Shock Wave , 1965 .

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

[20]  Alexandre Ern,et al.  Multicomponent transport algorithms , 1994 .

[21]  Giampiero Spiga,et al.  A note on the kinetic theory of chemically reacting gases , 1999 .

[22]  E. Isaacson,et al.  Numerical Analysis for Applied Science , 1997 .

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