Khrapov S.S., Kuzmin N.M., Butеnko M.A. The Comparison of Accuracy and Convergence for the CSPH – TVD Method and Some Eulerian Schemes for Solving Gas-Dynamic Equations
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https://doi.org/10.15688/jvolsu1.2016.6.15
Sеrgеy Sеrgееvich Khrapov
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Information Systems and Computer Simulation,
Volgograd State University
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,
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Nikolay Mikhaylovich Kuzmin
Candidate of Physical and Mathematical Sciences, Associate Professor,
Department of Information Systems and Computer Simulation,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Mariya Anatolyevna Butеnko
Senior Lecturer, Department of Information Systems and Computer Simulation,
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. 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 are presented. 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 usually used for gas-dynamic problems.
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 show 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.
Key words: numerical simulation, gas-dynamics, Lagrange — Eulerian approach, order of convergence, accuracy of numerical solution.
The Comparison of Accuracy and Convergence for the CSPH – TVD Method and Some Eulerian Schemes for Solving Gas-Dynamic Equations by Khrapov S.S., Kuzmin N.M., Butеnko M.A. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №6 (37) 2016 pp. 166-173
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