Geodesic
Numerical scheme cSPH --- TVD: investigation of influence slope limiters
Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2014), pp. 22-33

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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 $L_1$ relative 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.
Keywords: numerical schemes, SPH, TVD, slope limiters, combined lagrange-eulerian approach. 
@article{VVGUM_2014_1_a3,
     author = {N. M. Kuz'min and A. V. Belousov and T. S. Shushk{\cyre}vich and S. S. Khrapov},
     title = {Numerical scheme {cSPH} --- {TVD:} investigation of influence slope limiters},
     journal = {Matemati\v{c}eska\^a fizika i kompʹ\^uternoe modelirovanie},
     pages = {22--33},
     publisher = {mathdoc},
     number = {1},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VVGUM_2014_1_a3/}
}
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N. M. Kuz'min; A. V. Belousov; T. S. Shushkеvich; S. S. Khrapov. Numerical scheme cSPH --- TVD: investigation of influence slope limiters. Matematičeskaâ fizika i kompʹûternoe modelirovanie, no. 1 (2014), pp. 22-33. http://geodesic.mathdoc.fr/item/VVGUM_2014_1_a3/