This paper presents a review of the progress of smoothed particle hydrodynamics (SPH) towards high-order converged simulations. As a mesh-free Lagrangian method suitable for complex flows with interfaces and multiple phases, SPH has developed considerably in the past decade. While original applications were in astrophysics, early engineering applications showed the versatility and robustness of the method without emphasis on accuracy and convergence. The early method was of weakly compressible form resulting in noisy pressures due to spurious pressure waves. This was effectively removed in the incompressible (divergence-free) form which followed; since then the weakly compressible form has been advanced, reducing pressure noise. Now numerical convergence studies are standard. While the method is computationally demanding on conventional processors, it is well suited to parallel processing on massively parallel computing and graphics processing units. Applications are diverse and encompass wave–structure interaction, geophysical flows due to landslides, nuclear sludge flows, welding, gearbox flows and many others. In the state of the art, convergence is typically between the first- and second-order theoretical limits. Recent advances are improving convergence to fourth order (and higher) and these will also be outlined. This can be necessary to resolve multi-scale aspects of turbulent flow.

中文翻译：

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平滑粒子流体动力学综述：收敛拉格朗日流建模

本文回顾了平滑粒子流体动力学 (SPH) 在高阶收敛模拟方面的进展。作为一种适用于具有界面和多相的复杂流动的无网格拉格朗日方法，SPH 在过去十年中得到了长足的发展。虽然最初的应用是在天体物理学中，但早期的工程应用显示了该方法的多功能性和稳健性，而不强调准确性和收敛性。早期的方法是弱可压缩形式，由于虚假压力波导致噪音压力。这在随后的不可压缩（无发散）形式中被有效地去除了；从那时起，弱可压缩形式得到了改进，降低了压力噪音。现在数值收敛研究是标准的。虽然该方法对传统处理器的计算要求很高，它非常适合在大规模并行计算和图形处理单元上进行并行处理。应用多种多样，包括波浪-结构相互作用、滑坡引起的地球物理流、核污泥流、焊接、齿轮箱流等。在现有技术中，收敛通常在一阶和二阶理论极限之间。最近的进展正在改进对四阶（及更高阶）的收敛，这些也将被概述。这对于解决湍流的多尺度方面是必要的。收敛通常在一阶和二阶理论极限之间。最近的进展正在改进对四阶（及更高阶）的收敛，这些也将被概述。这对于解决湍流的多尺度方面是必要的。收敛通常在一阶和二阶理论极限之间。最近的进展正在改进对四阶（及更高阶）的收敛，这些也将被概述。这对于解决湍流的多尺度方面是必要的。