A class of second-derivatives in the Smoothed Particle Hydrodynamics with 2nd-order accuracy and its application to incompressible flow simulations

In this paper, we propose a class of decoupled first- and second-derivatives models for the Smoothed Particle Hydrodynamics (SPH) method, which were inspired by the Lagrangian Differencing Dynamics (LDD) (Bašić et al., 2018) and (Bašić et al., 2022) and arranged to the SPH framework. Being extensions of existing gradient and Laplacian SPH models, the proposed decoupled models include the cross-derivatives, which are crucial to ensure 2nd-order accuracy. Under the framework of the proposed class of decoupled models, we defined five second-derivative models, here termed, Naïve (NI), Block Diagonal (BD), SUM, Part Inverse (PI), and Full Inverse (FI). Among these models, only the NI and BD have equivalences in existing models, while the other three are completely original. Therefore, we believe that, besides the development of new models, this work may provide a standard classification for second-derivative models in SPH. In addition, applying the FI model into the first-derivative model, we propose the first instance of a 2nd-order accurate gradient model in SPH. Then, we demonstrated that the convergence rate of the truncation error decreases the higher the order of the model, and the results were investigated for different particle disorder levels. After that, we applied the proposed derivative models to incompressible fluid flow simulations for validation. The validation includes four famous benchmark tests: the lid-driven cavity, the Kármán vortex behind a cylinder, the rotating square patch, and the dam break problem. The first two tests are used to discuss the accuracy of the spatial derivative models without a free surface, while the third test already includes the effects of the free surface, where particle disorder is more severe. All examples demonstrate the robustness and accuracy of our proposed spatial derivatives at 2nd-order accuracy. Since the combination of the FI Laplacian model and the 2nd-order gradient model can satisfy 2nd-order accuracy, we expect this to be the second generation of SPH formulations, which we call SPH(2).