Conservative, density-based smoothed particle hydrodynamics with improved partition of the unity and better estimation of gradients

Abstract

Context. The accurate evaluation of gradients is a cornerstone of the smoothed particle hydrodynamics (SPH) technique. Using an integral approach to estimating gradients has been proven to substantially enhance its accuracy, retaining the Lagrangian structure of SPH equations and remaining fully conservative. However, in practice, it is difficult to ensure that the Lagrangian formulation is entirely consistent with regard to the exact partition of the unity. Aims. In this paper, we focus our study on the connection between the choice of the volume elements (VEs) in the SPH summations as well as the accuracy in the gradient estimation within the integral approach scheme (ISPH). We propose a new variant of VEs to improve the partition of the unity that is fully compatible with the Lagrangian formulation of SPH, including grad-h corrections. Aims. Using analytic considerations, simple static toy models in 1D, and a set of 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 handling sharp density contrasts. Methods. The ISPH code that is 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 shown to be excellent in all of these tests, consistently demonstrating that an improvement in the partition of the unity is not detrimental to the optimal conservation of energy, momentum, and entropy that is typical of Lagrangian schemes. Results. We successfully built a new ISPH scheme on a Lagrangian basis, which is fully conservative, and compatible with self-consistent grad-h terms and an improved partition of the unity. The ensuing code is able to successfully cope with the tensile instability and has been validated with a number of hydrodynamic tests with good results.