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An efficient discrete-velocity method for the Boltzmann equation
ComputerPhysics Communications Computer Physics Communications121-122 (1999) 717 www.elsevier.nl/locate/cpc
Abstract
An efficient discrete-velocity method for the Boltzmann equation T. Ptatkowski 1, W. Walu~ 2 b~stitute of Applied Mathematics and Mechanics, Departmentof Mathematics, Informatics and Mechanics, WarsawUniversity, Poland
The main difficulty in solving numerically the Boltzmann equation
Of Ot
~-V~x = J (f, f)
is contained in evaluation of the collision integral J (f, f ). The collision integral, in kinetic theory of rarefied gases, is a five-fold integral over an unbounded integration domain and depends nonlinearlyon the distribution function f = f ( t , x, o) (t stands for time, x for position and v for velocity). Early methods of evaluation of the collision integral were based mainly on Monte Carlo quadratures. In the last decade a number of new methods of numerical evaluation of the collision integral J(f, f ) were proposed. There is a class of such methods, often referred to as the discrete velocity methods, in which during the discretization process the collision integral J(f, f ) is replaced by a discrete collision term. These methods provide links between the Boltzmann equation and the discrete velocity models in kinetic theory. For one of such methods a number of important rigorous results concerning consistency and convergence where established. This method, while sound on a theoretical basis, in general has computational complexity at least of order O(N 2) (N be!ng the number of the discrete velocities taken in the approximation). In order to apply this method to solve real physical problems this complexity of the method must be substantially reduced. In this paper we present a simple modification of the discrete velocity algorithm for the spatially homogeneous Boltzmann equation which reduces significantly the computational complexity almost without lowering the accuracy. The modified algorithm is validated on a number of test problems for the spatially homogeneous Boltzmann equation. Moreover, the performance of this algorithm is also verified in solving the one-dimensional shock wave problem. © 1999 Elsevier Science B.V. All fights reserved.
1E-mail:[email protected]. 2 E-mail:[email protected]. 0010-4655/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved.
Abstract
An efficient discrete-velocity method for the Boltzmann equation T. Ptatkowski 1, W. Walu~ 2 b~stitute of Applied Mathematics and Mechanics, Departmentof Mathematics, Informatics and Mechanics, WarsawUniversity, Poland
The main difficulty in solving numerically the Boltzmann equation
Of Ot
~-V~x = J (f, f)
is contained in evaluation of the collision integral J (f, f ). The collision integral, in kinetic theory of rarefied gases, is a five-fold integral over an unbounded integration domain and depends nonlinearlyon the distribution function f = f ( t , x, o) (t stands for time, x for position and v for velocity). Early methods of evaluation of the collision integral were based mainly on Monte Carlo quadratures. In the last decade a number of new methods of numerical evaluation of the collision integral J(f, f ) were proposed. There is a class of such methods, often referred to as the discrete velocity methods, in which during the discretization process the collision integral J(f, f ) is replaced by a discrete collision term. These methods provide links between the Boltzmann equation and the discrete velocity models in kinetic theory. For one of such methods a number of important rigorous results concerning consistency and convergence where established. This method, while sound on a theoretical basis, in general has computational complexity at least of order O(N 2) (N be!ng the number of the discrete velocities taken in the approximation). In order to apply this method to solve real physical problems this complexity of the method must be substantially reduced. In this paper we present a simple modification of the discrete velocity algorithm for the spatially homogeneous Boltzmann equation which reduces significantly the computational complexity almost without lowering the accuracy. The modified algorithm is validated on a number of test problems for the spatially homogeneous Boltzmann equation. Moreover, the performance of this algorithm is also verified in solving the one-dimensional shock wave problem. © 1999 Elsevier Science B.V. All fights reserved.
1E-mail:[email protected]. 2 E-mail:[email protected]. 0010-4655/99/$ - see front matter © 1999 ElsevierScienceB.V. All rights reserved.