A generalized smoothed particle hydrodynamics method based on the moving least squares method and its discretization error estimation

Abstract

This paper demonstrates that the least squares method can generalize conventional discretized models of the smoothed particle hydrodynamics (SPH) method and proposes a simple discretization error evaluation for the SPH method, which is based on the truncation error of the least squares method. Since classical SPH models are formulated under the ideal assumption of a uniform particle distribution, their accuracies deteriorate to zeroth order or worse when the particle configuration is disordered. Although numerous advanced SPH models have been developed that ensure the spatial discretization accuracies of first order or higher under non-ideal conditions, their similarities and differences remain unexplored. This has motivated us to construct a generalized formulation encompassing existing SPH models. Almost all of the classical SPH and the advanced SPH models can be mathematically unified by the generalized particle method based on the least squares fitting, namely, the moving least squares (MLS) method. By deriving its truncation error, we analytically evaluate and numerically verify the discretization errors of various SPH models. These results confirm that all the classical SPH models exhibit zeroth-order or “negative” first-order accuracy, whose error increases as the particle spacing decreases. This paper proposes a generalized SPH model based on the MLS scheme with arbitrary accuracy for spatial derivatives of any order. This model is referred to as the least squares SPH (LSSPH) model. Additionally, we perform some benchmark tests to validate the LSSPH model with second-order accuracy for the zeroth and the first derivatives and first-order accuracy for the second derivatives.