Geodesic
A Shock Capturing Method
Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 1, pp. 62-75 Cet article a éte moissonné depuis la source Math-Net.Ru

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Strong discontinuities, or shocks in continua are a result of external dynamic loads. On the shock surface the conservation laws take the form of nonlinear algebraic equations for jumps across the shock. Entropy jumps across a strong discontinuity, and just this jump differs shocks from waves where the quantities vary continuously. In the heterogeneous difference schemes, the shock is treated as a layer of a finite thickness comparable with the cell size. This property of finite-difference schemes was called distraction. Since the state behind a shock is related to the state before it by the Hugoniot, in the distraction region there must act a mechanism that increases entropy. The physical viscosity and heat conductivity in continuum mechanics equations do not make it unnecessary to introduce a shock surface and hence cannot make the distraction length comparable with a few cells of the difference mesh. The paper considers a number of finite difference schemes where energy dissipation in the distraction region is defined by equations which are valid on the shock surface.
Keywords: shock wave; differential method; distraction; energy dissipation; conservation laws. 
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V. F. Kuropatenko. A Shock Capturing Method. Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematičeskoe modelirovanie i programmirovanie, Tome 7 (2014) no. 1, pp. 62-75. http://geodesic.mathdoc.fr/item/VYURU_2014_7_1_a5/

[1] Kuropatenko V. F., “Finite Difference Methods for Hydrodynamics Equations”, Proceedings of the Steklov Institute of Mathematics, 74, no. 1, 1966, 107–137 | MR | Zbl

[2] J. Neumann, R. Richtmayer, “A Method for the Numerical Calculation of Hydrodynamical Shocks”, J. Appl. Phys., 21:3 (1950), 232–237 | DOI | MR | Zbl

[3] P. D. Lax, “Weak Solution of Nonlinear Hyperbolic Equations and Their Numerical Computations”, Comn. Pure and Appl. Math., 7 (1954), 159–193 | DOI | MR | Zbl

[4] Godunov S. K., “A Finite-Difference Method for Shock Calculation”, Russian Mathematical Surveys, 12:1 (1957), 176–177 | MR | Zbl

[5] Kuropatenko V. F., “A Shock Calculation Method”, DAN SSSR, 3:4 (1960), 771–772 | MR

[6] Rohzdestvensky B. L., Yanenko N. N., Systems of Quasi-Linear Equations and Their Applications to Hydrodynamics, Nauka, M., 1968, 592 pp. | MR

[7] Kuropatenko V. F., Makeyeva I. R., “Discontinuity Distraction in Shock Calculation Methods”, Mathematical Models and Computer Simulations, 18:3 (2006), 120–128 | MR | Zbl

[8] Stupochenko E. V., Losev S. A., Osipov A. I., Relaxation Processes in Shock Waves, Nauka, M., 1965, 484 pp.

[9] Kuropatenko V. F., Makeyeva I. R., A Higher-Monotonicity Finite-Difference Shock Capture Method, VNIITF Preprint, No 120, 1997 | MR

[10] Kuropatenko V. F., “Local Conservatism of Difference Schemes for Hydrodynamics Equations”, Computational Mathematics and Mathematical Physics, 25:8 (1985), 1176–1188 | MR

[11] Kuropatenko V. F., “Ultimate Conservatism of Finite-Difference Conservation Laws”, Atomic Science and Engineering. Series: Numerical Methods of Mathematical Physics, 1982, no. 3(11), 3–5

[12] Kuropatenko V. F., “Entropy Accuracy in Finite Difference Schemes for Hydrodynamics Equations”, Numerical Methods for Continuum Mechanics, 9:7 (1978), 49–59 | MR

[13] Kuropatenko V. F., “Divergence and Conservatism of Finite-Difference Schemes for Hydrodynamics Equations”, Atomic Science and Engineering. Series: Mathematical Modeling of Physical Processes, 1990, no. 2, 63–69

[14] Kuropatenko V. F., “Shock Calculation Methods”, Mathematical Modelling of Low-Temperature Plasma, v. 2, Encyclopedia of Low-Temperature Plasma. Series B, VII-I, 2008, 496–506

[15] Kuropatenko V. F., Continuum mechanics models, Chelyabinsk State University, Chelyabinsk, 2007, 302 pp.

[16] Kuropatenko V. F., Dorovskikh I. A., Makeyeva I. R., “The Properties of Finite Difference Schemes and Simulation of Dynamic Processes”, Computating Technologies, 11:2 (2006), 9–11

[17] Kuropatenko V. F., Kovalenko G. V., Kuznetsova V. I., Mikhaylova G. I., “Complex Programs VOLNA and Method for Transient Flows of Continua”, Atomic Science and Engineering, Series: Mathematical Modeling of Physical Processes, 1989, no. 2, 9–25