Two variants of Monte Carlo projection method for numerical solution of nonlinear Boltzmann equation
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Sergey V. Rogasinsky
Abstract
The paper is focused on justification of a statistical modelling algorithm for solution of the nonlinear kinetic Boltzmann equation on the base of a projection method. Hermite functions are used as an orthonormal basis. The error of approximation of a function by a partial sum of Hermite functions series is estimated in the L2 norm. The estimates are compared for two variants of the projection method in the case of solutions to the homogeneous gas relaxation problem with a known solution.
Funding: The work was supported by the Russian Foundation for Basic Research (project No. 18-01-00356 a), and the Program of fundamental research of the Presidium of RAS I.33II.
References
[1] A. V. Bobylev, Exact solutions to Boltzmann equation. Doklady Akad. Nauk SSSR225 (1975), No. 6, 1296–1299.Search in Google Scholar
[2] N. N. Chentsov, Statistical Decision Rules and Optimal Conclusions. Nauka, Moscow, 1972 (in Russian).Search in Google Scholar
[3] H. Grad, On the kinetic theory of rarefied gases. Comm. Pure Appl. Math. 2 (1949), 331–407.10.1002/cpa.3160020403Search in Google Scholar
[4] M. S. Ivanov and S. V. Rogasinsky, Economical schemes of statistical modelling of flows of rarefied gas. Matem. Model. 1 (1989), 130–145.Search in Google Scholar
[5] D. Jackson, Fourier Series and Orthogonal Polynomials. Amer. Math. Soc., 1941.10.5948/UPO9781614440062Search in Google Scholar
[6] M. Kac, Probability and Related Topics in Physical Sciences. Amer. Math. Soc., 1959.Search in Google Scholar
[7] M. N. Kogan, Rarefied Gas Dynamics. Nauka, Moscow, 1967 (in Russian).Search in Google Scholar
[8] A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis. Nauka, Moscow, 2004 (in Russian).Search in Google Scholar
[9] G. A. Mikhailov and A. V. Voitishek, Numerical Statistical Modelling. Monte Carlo Methods. Academia Publ. Moscow, 2006 (in Russian).Search in Google Scholar
[10] S. V. Rogazinsky, Statistical modelling algorithm for solving the nonlinear Boltzmann equation based on the projection method. Russ. J. Numer. Anal. Math. Modelling32 (2017), No. 3, 197–202.10.1515/rnam-2017-0017Search in Google Scholar
[11] P. K. Suetin, Classic Orthogonal Polynomials. Fizmatlit, Moscow, 2005 (in Russian).Search in Google Scholar
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Articles in the same Issue
- Frontmatter
- Efficient projection method for a system of differential equations of Fokker−Planck type
- Two variants of Monte Carlo projection method for numerical solution of nonlinear Boltzmann equation
- Subgrid modelling of convective diffusion in a multiscale random medium
- Solution method for underdetermined systems of nonlinear equations
- Boundary least squares method with three-dimensional harmonic basis of higher order for solving linear div-curl systems with Dirichlet conditions
Articles in the same Issue
- Frontmatter
- Efficient projection method for a system of differential equations of Fokker−Planck type
- Two variants of Monte Carlo projection method for numerical solution of nonlinear Boltzmann equation
- Subgrid modelling of convective diffusion in a multiscale random medium
- Solution method for underdetermined systems of nonlinear equations
- Boundary least squares method with three-dimensional harmonic basis of higher order for solving linear div-curl systems with Dirichlet conditions
