Kuzmin N.M., Belousov A.V., Shushkevich T.S., Khrapov S.S. Numerical scheme CSPH – TVD: investigation of Influence slope limiters

 
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
 
Shushkеvich Tat’yana Sеrgееvna
 
Student, Institute of Mathematics and IT
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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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.
 
Key words: numerical schemes, SPH, TVD, slope limiters, combined lagrange-eulerian approach.
 

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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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