In this paper a pseudo-spectral incompressible smoothed particle hydrodynamics (FFT-ISPH) solver is presented. While the solution of the linear system arising from the pressure Poisson equation in physical space using iterative solvers in incompressible SPH is a viable solution, it is widely accepted that the computational cost of solving the pressure Poisson equation by iterative solvers is excessive and affects negatively on the efficiency of the solver. The proposed scheme is an intermediate between a fully spectral and a standard incompressible SPH solver. Herein, the solution of the pressure Poisson equation is performed in spectral space whereas the discretisation and time integration are performed in physical space. This results in a second and higher-order scheme with the classical first-order projection and a third-order Runge-Kutta time integration scheme, respectively. A detailed performance analysis shows gains in computational cost of two orders of magnitude. Further, it is demonstrated that the solution of the pressure Poisson equation in spectral space is independent of the number of neighbouring nodes, in contrast to the iterative solver whose cost increases by a factor of three with smoothing length to particle size ratio. Periodic, bounded, and mixed boundary conditions test cases have been used to demonstrate the applicability, accuracy and robustness of the scheme with second and fourth-order convergence rates. The scheme opens an avenue for simulations of high-order incompressible flows in SPH where filtering operations in the frequency domain can be performed straightforwardly.

本文提出了一种伪谱不可压缩平滑粒子流体动力学（FFT-ISPH）求解器。尽管使用不可压缩SPH中的迭代求解器求解物理空间中的压力泊松方程所产生的线性系统是可行的解决方案，但已被广泛接受的是，使用迭代求解器求解压力泊松方程的计算成本过高，并且对系统产生负面影响。求解器的效率。所提出的方案是全光谱和标准不可压缩SPH求解器之间的中间方案。在此，压力泊松方程的解在频谱空间中执行，而离散化和时间积分在物理空间中执行。这导致分别具有经典一阶投影和第二阶Runge-Kutta时间积分方案的第二阶和更高阶方案。详细的性能分析显示，计算成本增加了两个数量级。此外，与迭代求解器相比，证明了其在泊松空间中的压力泊松方程的求解与相邻节点的数量无关，而迭代求解器的成本以平滑长度与粒径之比增加了三倍。周期性，有界和混合边界条件测试用例已被用来证明该方案具有二阶和四阶收敛速度的适用性，准确性和鲁棒性。