An SPH scheme based on targeted essentially nonoscillatory reconstruction and its applications

Abstract

<p indent="0mm">In this paper, we developed an SPH scheme based on targeted essentially nonoscillatory (TENO) reconstruction. In this scheme, the Riemann-SPH method is applied to solve governing equations. However, the original Riemann-SPH generates excessive numerical dissipations due to the adoption of an approximate Riemann solver, thus exhibiting low numerical accuracy for simulating fluid flows. To solve this problem, we applied a five-point TENO method to reconstruct the left and right states of the Riemann problem. In the original TENO method, five equidistant stencil points are required. However, due to the nature of the Lagrangian-SPH method, the particle distribution is typically random. It is difficult for SPH to find equidistant points. To mitigate this issue, we first consider a particle pair as two stencil points and find the particle closest to the missing point. Through the gradient approximation, we can obtain the primitive values of the missing point. Based on these values, we can implement the TENO reconstruction smoothly. The excessive numerical dissipations are successfully reduced through TENO reconstruction, and the numerical accuracy is further improved. Importantly, the proposed TENO-SPH scheme avoids using the artificial viscosity commonly used in the conventional SPH. Moreover, TENO-SPH has a robust numerical ability, and thus, it can be applied to reproduce typical compressible flows, vortex flows, and free surface flows with superior accuracy.