An Energy Stable SPH Method for Incompressible Fluid Flow

In this paper, a novel unconditionally energy stable Smoothed Particle Hydrodynamics (SPH) method is proposed and implemented for incompressible fluid flows. In this method, we apply operator splitting to break the momentum equation into equations involving the non-pressure term and pressure term separately. The idea behind the splitting is to simplify the calculation while still maintaining energy stability, and the resulted algorithm, a type of improved pressure correction scheme, is both efficient and energy stable. We show in detail that energy stability is preserved at each full-time step, ensuring unconditionally numerical stability. Numerical examples are presented and compared to the analytical solutions, suggesting that the proposed method has better accuracy and stability. Moreover, we observe that if we are interested in steady-state solutions only, our method has good performance as it can achieve the steady-state solutions rapidly and accurately.