Conservative, density-based smoothed particle hydrodynamics with improved partition of the unity and better estimation of gradients. (arXiv:2101.07364v3 [astro-ph.IM] UPDATED)
<a href="http://arxiv.org/find/astro-ph/1/au:+Garcia_Senz_D/0/1/0/all/0/1">Domingo Garc&#xed;a-Senz</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Cabezon_R/0/1/0/all/0/1">Rub&#xe9;n M. Cabez&#xf3;n</a>, <a href="http://arxiv.org/find/astro-ph/1/au:+Escartin_J/0/1/0/all/0/1">Jos&#xe9; A. Escart&#xed;n</a>

The correct evaluation of gradients is at the cornerstone of the smoothed
particle hydrodynamics (SPH) technique. Using an integral approach to estimate
gradients has proven to enhance accuracy substantially. Such approach retains
the Lagrangian structure of SPH equations and is fully conservative. But, in
practice, it is difficult to make the Lagrangian formulation totally consistent
to an exact partition of the unity.

In this paper we study, among other things, the connection between the choice
of the volume elements (VEs), which enters in the SPH summations, and the
accuracy in the gradient estimation within the integral approach scheme (ISPH).
A new variant of VEs are proposed which improve the partition of the unity and
is fully compatible with the Lagrangian formulation of SPH, including the
grad-h corrections. Using analytic considerations, simple static toy models in
1D, and a few full 3D test cases, we show that any improvement in the partition
of the unity also leads to a better calculation of gradients when the integral
approach is used jointly. Additionally, we propose an easy-to-implement
modification of the ISPH scheme, which makes it more flexible and better suited
to handle sharp density contrasts.

The ISPH code built with the proposed scheme has been validated with a good
number of standard tests, some of them involving contact discontinuities. The
performance of the code was excellent in all of them, showing that an
improvement in the partition of the unity is not detrimental of the good
conservation of energy, momentum, and entropy typical of Lagrangian schemes.

The correct evaluation of gradients is at the cornerstone of the smoothed
particle hydrodynamics (SPH) technique. Using an integral approach to estimate
gradients has proven to enhance accuracy substantially. Such approach retains
the Lagrangian structure of SPH equations and is fully conservative. But, in
practice, it is difficult to make the Lagrangian formulation totally consistent
to an exact partition of the unity.

In this paper we study, among other things, the connection between the choice
of the volume elements (VEs), which enters in the SPH summations, and the
accuracy in the gradient estimation within the integral approach scheme (ISPH).
A new variant of VEs are proposed which improve the partition of the unity and
is fully compatible with the Lagrangian formulation of SPH, including the
grad-h corrections. Using analytic considerations, simple static toy models in
1D, and a few full 3D test cases, we show that any improvement in the partition
of the unity also leads to a better calculation of gradients when the integral
approach is used jointly. Additionally, we propose an easy-to-implement
modification of the ISPH scheme, which makes it more flexible and better suited
to handle sharp density contrasts.

The ISPH code built with the proposed scheme has been validated with a good
number of standard tests, some of them involving contact discontinuities. The
performance of the code was excellent in all of them, showing that an
improvement in the partition of the unity is not detrimental of the good
conservation of energy, momentum, and entropy typical of Lagrangian schemes.

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