A Method for Studying the Dissipative Properties of Difference Schemes in Euler Coordinates

At present, numerical methods for calculating shock-wave flows of liquid and gas in Eulerian coordinates have become widespread; therefore, the study of their characteristics is an urgent task. The paper presents an approach to assessing the dissipative properties of such difference schemes on strong discontinuities. The idea of the method is to construct the entropy production equation, the approximation error of which can be expressed by a combination of the approximation errors of the equations that make up the difference scheme. The equation of entropy production on a weak shock wave is used as a criterion for the dissipativity of the difference scheme. The paper evaluates the dissipative properties of the large particle method using the proposed method.

Keywords

difference schemes; Euler coordinates; dissipativity; entropy production.

References

1. Yanenko N.N., Shokin Y.I. The First Differential Approximation to Finite-Difference Schemes for Hyperbolic Systems of Equations. Siberian Mathematical Journal, 1969, vol. 10, no. 5, pp. 868-880. DOI: 10.1007/BF00971662
2. Rozhdestvenskii B.L., Yanenko N.N. Sistemy kvazilineinykh uravnenii [Systems of Quasilinear Equations]. Мoscow, Nauka, 1968.
3. Zel'dovich YA.B., Raizer YU.P. Fizika udarnykh voln [Physics of Shock Waves]. Moscow, Fizmatgiz, 1963.
4. Zababakhin E.I. Nekotorye voprosy gazodinamiki vzryva [Some Questions of the Gas Dynamics of the Explosion]. Snezhinsk, RFYaTs-VNIITF, 1997.
5. Kuropatenko V.F., Shestakovskaya E.S. Osnovy chislennykh metodov mekhaniki sploshnoi sredy [Fundamentals of Numerical Methods of Continuum Mechanics]. Chelyabinsk, Publisher Center of South Ural State University, 2017.
6. Kuropatenko V.F. Local Conservativeness of Difference Schemes for the Equations of Gas Dynamics. Computational Mathematics and Mathematical Physics, 1985, vol. 25, no. 8, pp. 1176-1188. DOI: 10.1016/0041-5553(85)90157-0
7. Belotserkovskii S.M., Davydov YU.M. Metod krupnykh chastits v gazovoi dinamike [The Method of Large Particles in Gas Dynamics]. Moscow, Nauka, 1982.