Kuzmin N.M., Belousov A.V., Shushkevich T.S., Khrapov S.S. Numerical scheme CSPH – TVD: investigation of Influence slope limiters
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Kuz’min Nikolay Mikhaylovich
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
Bеlousov Anton Vladimirovich
Student, Institute of Mathematics and IT
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Shushkеvich Tat’yana Sеrgееvna
Student, Institute of Mathematics and IT
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Khrapov Sеrgеy Sеrgееvich
Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Information Systems and Computer Simulation
Volgograd State University
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Volgograd State University
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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation
Abstract. The generalisation of combined lagrange-eulerian numerical scheme cSPH — TVD for ideal gas-dynamics equations without extarnal forces in one-dimensional case was described. The results of the Riemann problems numerical simulation for different variants of this numerical scheme are shown. Influence of slope-limitiers and flux computation methods to quality of numerical solution are investigated.
Six version of slope limiters are investigated: minmod, van Leer, van Albada, Kolgan, k-parameter and Colella — Woodward. Two methods of numerical flux computation also investigated: Lax — Friedrichs and Harten — Lax — van Leer.
It is shown, that two pair of slope limiters leads to very similar numerical solution quality: minmod — Kolgan and van Leer — Colella — Woodward for the both version of numerical flux computation — Lax — Friedrichs and Harten — Lax — van Leer methods.
For the Lax — Friedrichs method of numerical flux computation Colella–Woodward slope limiter give the best results and minmod the worse. For the Harten — Lax — van Leer method of numerical flux computation k-parameter slope limiter give the best results and Kolgan the worse.
The L1relative error in density varying from 1.76% to 3.1% depending on the numerical flux computation method and kind of slope limiter.
It is shown, that for all investigated variants of cSPH — TVD method numerical solution of Riemann problem very similar to exact.
It is very interesting, that k-parameter slope limiter in combination with Lax — Friedrichs method of numerical flux computation leads to strange features near to contact discontinuity and rarefaction wave. But, in combination with Harten — Lax — van Leer method of numerical flux computation it leads to the best of all results without these strange features.
It is shown, that two pair of slope limiters leads to very similar numerical solution quality: minmod — Kolgan and van Leer — Colella — Woodward for the both version of numerical flux computation — Lax — Friedrichs and Harten — Lax — van Leer methods.
For the Lax — Friedrichs method of numerical flux computation Colella–Woodward slope limiter give the best results and minmod the worse. For the Harten — Lax — van Leer method of numerical flux computation k-parameter slope limiter give the best results and Kolgan the worse.
The L1relative error in density varying from 1.76% to 3.1% depending on the numerical flux computation method and kind of slope limiter.
It is shown, that for all investigated variants of cSPH — TVD method numerical solution of Riemann problem very similar to exact.
It is very interesting, that k-parameter slope limiter in combination with Lax — Friedrichs method of numerical flux computation leads to strange features near to contact discontinuity and rarefaction wave. But, in combination with Harten — Lax — van Leer method of numerical flux computation it leads to the best of all results without these strange features.
Key words: numerical schemes, SPH, TVD, slope limiters, combined lagrange-eulerian approach.
Numerical Scheme CSPH – TVD: Investigation of Influence Slope Limiters by Kuzmin N.M., Belousov A.V., Shushkevich T.S., Khrapov S.S. is licensed under a Creative Commons Attribution 4.0 International License.
Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №1 (20) 2014 pp. 22-34
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