A new type of WENO scheme in SPH for compressible flows with discontinuities

Abstract

In the past few decades, the weighted essentially non-oscillatory (WENO) scheme has been widely used in mesh-based methods and has achieved great success, but its application to the smoothed particle hydrodynamics (SPH) is still limited. In this paper, a simple and accurate implementation of the WENO scheme to the SPH method is proposed for solving compressible flows with discontinuities and small-scale structures. In the proposed scheme, several equal spacing points along the line joining two interacting particles are selected to constitute candidate stencils. However, due to the Lagrangian characteristic of SPH, some points may be lost. To solve this problem, we first search the particles closest to these missing points, and then the first-order Taylor series expansion is used to obtain the corresponding primitive variables of these points. After that, through the WENO strategy, the left and right states at the interface between two interacting particles are reconstructed. Finally, the inter-particle interactions are determined by using Roe’s approximate Riemann solver. Several numerical tests show that the proposed WENO-SPH method is robust and able to accurately capture shockwaves, and benefiting from the low-dissipation property, it also has a good performance in resolving small-scale structures in flows.