Godunov's Method for a Multivelocity Model of Heterogeneous Medium

This article uses a model of heterogeneous media accounting for an additional state of the medium as a mixture characterized by averaged quantities. The equations describing this state coincide with the equations of gas dynamics. Additional equations express conservation laws, but only for the components with the local speed of sound lower than in the mixture; we assume that other waves are absorbed by the media and form waves in the mixture. Since the equations are not in divergence form, the original Godunov's method is inapplicable. We suggest a modified Godunov's method to integrate the nondivergent system of equations for a multivelocity heterogeneous mixture. We use a linearized Riemann solver for Riemann problems.

Keywords

multivelocity multicomponent medium; hyperbolic systems of PDEs not in divergence form; modification of Godunov's approach; linearized Riemann solver; numerical modelling.

References

1. Stewart H., Wendroff B. Two-Phase Flow: Models and Methods. Journal Comput. Phys., 1984, vol. 56, pp. 363-409. DOI: 10.1016/0021-9991(84)90103-7
2. Surov V.S. Hyperbolic Models in the Mechanics of Heterogeneous Media. Computational Mathematics and Mathematical Physics, 2014, vol. 54, no. 1, pp. 148-157. DOI: 10.1134/S096554251401014X
3. Surov V.S. Hyperbolic Model of a Multivelocity Heterogeneous Medium. Journal of Engineering Physics and Thermophysics, 2012, vol. 85, no. 3, pp. 530-538. DOI: 10.1007/s10891-012-0683-0
4. Godunov S.K., Zabrodin A.V., Ivanov M.Ya. Chislennoe reshenie mnogomernyih zadach gazovoy dinamiki [The Numerical Solution of Gas Dynamic Multidimensional Problems]. Moscow, Nauka, 1976.
5. Surov V.S. A Modification of Godunov Approach for Calculating Single Speed Flows of Multicomponent Mixtures. Matematicheskoe modelirovanie [Mathematical Modelling], 1998, vol. 10, no. 3, pp. 29-38. (in Russian)
6. Surov V.S. Breakup of Arbitrary Discontinuity in One-Velocity Heterogeneous Mixture of Compressible Media. High Temperature, 1998, vol. 36, no. 1, pp. 156-159.
7. Surov V.S. Toward the Calculation of Flows of a One-Velocity Multicomponent Mixture by the Modified S.K. Godunov Method. Journal of Engineering Physics and Thermophysics, 2011, vol. 84, no. 4, pp. 840-848. DOI: 10.1007/s10891-011-0541-5
8. Surov V.S. The Busemann Flow for a One-Velocity Model of a Heterogeneous Medium. Journal of Engineering Physics and Thermophysics, 2007, vol. 80, no. 4, pp. 681-688. DOI: 10.1007/s10891-007-0092-y
9. Surov V.S. The Riemann Problem for One-Velocity Model of Multicomponent Mixture. High Temperature, 2009, vol. 47, no. 2, pp. 263-271. DOI: 10.1134/S0018151X09020175
10. Surov V.S. On a Method of Approximate Solution of the Riemann Problem for a One-Velocity Flow of a Multicomponent Mixture. Journal of Engineering Physics and Thermophysics, 2010, vol. 83, no. 2, pp. 373-379. DOI: 10.1007/s10891-010-0354-y
11. Uollis G. Odnomernyie dvuhfaznyie techeniya [One Dimensional Two-Phase Flows]. Moscow, Mir, 1972.
12. Surov V.S. The Riemann Problem for the Multivelocity Model of a Multicomponent Medium. Journal of Engineering Physics and Thermophysics, 2013, vol. 86, no. 4, pp. 926-934. DOI: 10.1007/s10891-013-0913-0
13. Kulikovskii A.G., Pogorelov N.V., Semenov A.Yu. Matematicheskie voprosyi chislennogo resheniya giperbolicheskih sistem uravneniy [The Mathematical Complexities of Numerical Solving the Hyperbolic Systems]. Moscow, Fizmatlit, 2001.
14. Toro E.F. Riemann Solvers and Numerical Methods for Fluid Dynamics. Berlin, Springer, 1999. DOI: 10.1007/978-3-662-03915-1
15. Toro E.F. Riemann Solvers with Evolved Initial Condition. Int. Journal for Numerical Methods in Fluids, 2006, vol. 52, pp. 433-453. DOI: 10.1002/fld.1186