A novel high-order spectral incompressible smoothed particle hydrodynamics(ISPH) scheme with an immersed boundary method (IBM)

In this paper, a novel spectral Smoothed Particle Hydrodynamics (SPH) scheme is presented for solving incompressible flows. The implementation details, accuracy improvement and computational gains brought by spectral SPH scheme are investigated. It is shown that 4th and 6th order of convergence can be achieved for the differentiation operators through Gaussian type kernel functions in the spectral space when a sufficiently large kernel support is used. The pressure Poisson equation arising from the incompressible scheme is solved by the spectral equivalence which allows flexibility in different boundary conditions. Combination of the scheme with immersed boundary method (IBM) is explored. Compared to a direct forcing approach based on forcing spreading, a radial basis function (RBF) based IBM with the proposed scheme is able to provide an oscillation-free solution for simulating wall boundary conditions. The proposed spectral SPH scheme achieves a high order of convergence, as demonstrated through numerical tests. It is straightforward to implement using standard FFT libraries, such as FFTW3, and offers significant reductions in computational cost due to its elementwise operations and one-time FFT of kernel derivatives. These features make the scheme both practical and efficient for large-scale simulations. In this work, the spectral ISPH scheme is developed within an Eulerian framework, yet it possesses the potential to be naturally extended to a Lagrangian formulation, paving the way for simulating moving boundaries and complex flow dynamics.