An augmented HLLE approximate Riemann solver for the solution of weakly compressible SPH models

This paper presents a new formulation using an augmented Harten, Lax, van Leer, and Einfeldt (AHLLE) approximate Riemann solver for the solution of weakly compressible smoothed particle hydrodynamics (WCSPH). The proposed solver uses the standard scheme for the middle-state pressure term whilst introducing a novel formula for determining the middle-state velocity regime. This new formula is based on a wave propagation algorithm utilised to solve the 1D-Riemann problem governing neighbouring particle interactions. The differences between the velocities of the left and right states in the Riemann problem are treated as a series of waves propagating with the well-established and less dissipative Einfeldt speed. To mitigate associated dissipation, the AHLLE solver employs a wave-propagation algorithm in which the modified Monotonised Centre (MC) limiter is applied in a finite-volume manner to control the waves. Unlike many existing Riemann-SPH approaches, no quasi-conservative reformulation is required, enabling the method to preserve physical wave behaviour while avoiding excessive numerical damping. Furthermore, the method incorporates a particle-shifting technique (PST) to regulate particle distances. The proposed approach is then validated against standard and state-of-the-art SPH benchmarks, and the obtained results are compared against available analytical and experimental data, as well as existing SPH solvers. These comparisons affirm the effectiveness of the proposed method, demonstrating that pressure-acoustic noises are effectively damped without the need for artificial viscosity, while also avoiding the long-distance free-surface dissipation that affects many existing SPH formulations.