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Applied mathematics
Numerical scheme cSPH — TVD: investigation of influence slope limiters
N. M. Kuz'min,
A. V. Belousov,
T. S. Shushkåvich,
S. S. Khrapov Volgograd State University
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
$L_1$ 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.
UDC:
524.7-8
BBK:
22.193