Geodesic
The comparison of accuracy and convergence for the CSPH---TVD method and some Eulerian schemes for solving gas-dynamic equations
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 166-173.

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In this article the results of the study of accuracy and convergence methods CSPH—TVD, MUSCL, PPM and WENO for solving equations of ideal gas dynamics was present. Many physical processes are described by the equations of gas dynamics. Because of their non-linearity, exact or approximate analytical solutions can be obtained only for a limited number of special cases. Therefore, numerical methods are used for gas-dynamic problems usually. One of these methods is the numerical scheme CSPH—TVD (Combined Smooth Particle Hydrodynamics—Total Variation Diminishing). This method is based on the consistent application of the Lagrange (SPH) and Euler (TVD) approaches. Our results show that the relative error and orders the convergence of all the above numerical schemes are quite close. This can be due to the presence of a weak discontinuity in the structure of the solution. There are many solutions in gas dynamics problems with such discontinuities. The presence of a weak discontinuity allows us to understand the properties of numerical schemes in a real (not test) computations. Our results shows that CSPH—TVD method has comparable to popular Euler numerical schemes accuracy and convergence. A distinctive feature of this method is the possibility of computations at the boundary with vacuum (or dry bed for shallow water case). The additional regularization is not needed. The method is successfully used for solving various gas-dynamic problems in a variety of subject areas: the dynamics of surface water, aspiration flows, astrophysical jets, accretion flows, the transfer of contaminants and others.
Keywords: numerical simulation, gas-dynamics, Lagrange–Eulerian approach, order of convergence, accuracy of numerical solution. 
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     title = {The comparison of accuracy and convergence for the {CSPH---TVD} method and some {Eulerian} schemes for solving gas-dynamic equations},
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S. S. Khrapov; N. M. Kuz'min; M. A. Butenko. The comparison of accuracy and convergence for the CSPH---TVD method and some Eulerian schemes for solving gas-dynamic equations. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 6 (2016), pp. 166-173. http://geodesic.mathdoc.fr/item/VVGUM_2016_6_a15/

[1] A.\;G. Zhumaliev, S.\;S. Khrapov, “Numerical Scheme CSPH—TVD: Front of Shock Wave Simulation”, Science Journal of Volgograd State University. Mathematics. Physics, 2012, no. 2 (17), 60–67

[2] A.\;V. Khoperskov, S.\;S. Khrapov, A.\;V. Pisarev, A.\;A. Voronin, M.\;V. Eliseeva, I.\;A. Kobelev, “The Problem of Management Hydrological Regime in Ecology-Economic System «Volga HPS—Volga–Akhtuba Flood-Plain». Part 1. Simulation of Surface Water Dynamics in Springtime Flood”, Control Sciences, 2012, no. 5, 18–25

[3] A.\;A. Belosludtsev, D.\;V. Gusarov, M.\;A. Eremin, N.\;M. Kuzmin, S.\;A. Khoperskov, S.\;S. Khrapov, “Informational-Computer System for Modeling the Dynamics of Contaminants From the Chemical Industry”, Science Journal of Volgograd State University. Mathematics. Physics, 2009, no. 13, 95–102

[4] N.\;M. Kuzmin, V.\;V. Mustsevoy, S.\;S. Khrapov, “Numerical Simulation of Evolution of Unstable Modes of Jets From Young Star Objects”, Astronomy Reports, 84:12 (2007), 1089–1098

[5] N.\;M. Kuzmin, M.\;A. Butenko, “Numerical Code for Simulation of Aspiration Flows in Industry Workshop”, Science Journal of Volgograd State University. Mathematics. Physics, 2015, no. 5 (30), 51–59

[6] A.\;V. Pisarev, S.\;S. Khrapov, A.\;V. Khoperskov, “Numerical Scheme on the Base of Combined SPH—TVD Approach: the Problem of Shear Flows Simulation”, Science Journal of Volgograd State University. Mathematics. Physics, 2011, no. 2 (15), 138–141 | MR

[7] A.\;A. Voronin, A.\;A. Vasilchenko, M.\;V. Pisareva, A.\;V. Pisarev, A.\;V. Khoperskov, S.\;S. Khrapov, Yu.\;V. Podshchipkova, “Systems Design of Ecological and Economic Management of the Volga–Akhtuba Floodplain Based on Hydrodynamic and Geoinformatics Simulation”, Upravlenie bolshimi sistemami, 55, 2015, 79–102

[8] S.\;S. Khrapov, A.\;V. Khoperskov, M.\;A. Eremin, Simulation of Surface Water Dynamics, Izd-vo VolGU, Volgograd, 2010, 132 pp.

[9] A.\;V. Pisarev, S.\;S. Khrapov, E.\;O. Agafonnikova, A.\;V. Khoperskov, “Numerical Model of Surface Water Dynamics in Volgas Bed: Estimation of Roughness Coefficient”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2013, no. 1, 114–130

[10] S.\;S. Khrapov, A.\;V. Khoperskov, N.\;M. Kuzmin, A.\;V. Pisarev, I.\;A. Kobelev, “Numerical Scheme for Simulation of Dynamics Surface Waters on the Base of Combined SPH—TVD Approach”, Numerical Methods and Programming, 12:1 (2011), 282–297

[11] N.\;M. Kuzmin, A.\;V. Belousov, T.\;S. Shushkevich, S.\;S. Khrapov, “Numerical Scheme CSPH—TVD: Investigation of Influence Slope Limiters”, Science Journal of Volgograd State University. Mathematics. Physics, 2014, no. 1 (20), 22–34

[12] K.\;S. Shushkevich, N.\;M. Kuzmin, “One-Dimensional Numerical Scheme for Gas-Dynamics Simulation on the Base of Combined SPH—PPM Approach”, Vestnik magistratury, 2013, no. 5 (20), 40–44

[13] P. Colella, P.\;R. Woodward, “The piecewise parabolic method (PPM) for gas-dynamical simulations”, Journal of Computational Physics, 54:1 (1984), 174–201 | DOI | MR | Zbl

[14] A. Harten, P. Lax, B. van Leer, “On upstream differencing and Godunov type methods for hyperbolic conservation laws”, SIAM Review, 25:1 (1983), 35–61 | DOI | MR | Zbl

[15] G.-S. Jiang, C.\;W. Shu, “Efficient implementation of weighted ENO schemes”, Journal of Computational Physics, 126:1 (1996), 202–228 | DOI | MR | Zbl

[16] P.\;D. Lax, “Weak solutions of nonlinear hyperbolic equations and their numerical computation”, Communications on Pure and Applied Mathematics, 7:1 (1954), 159–193 | DOI | MR | Zbl

[17] B. van Leer, “Towards the ultimative conservative difference scheme. III. Upstreamcentered finite-difference schemes for ideal compressible flow”, Journal of Computational Physics, 23:3 (1977), 263–275 | DOI | MR | Zbl

[18] B. van Leer, “Towards the ultimate conservative difference scheme. V. A second order sequel to Godunov's method”, Journal of Computational Physics, 32:1 (1979), 101–136 | DOI | MR

[19] S.\;S. Khrapov, A.\;V. Pisarev, I.\;A. Kobelev, A.\;G. Zhumaliev, E.\;O. Agafonnikova, A.\;G. Losev, A.\;V. Khoperskov, “The numerical simulation of shallow water: estimation of the roughness coefficient on the flood stage”, Advances in Mechanical Engineering, 2013 (2013), 787016 | DOI