A New SPH Equation Including Variable Smoothing Lengths Aspects and Its Implementation

New SPH equations including fully variable smoothing length aspects and its implementation are proposed in this paper. Unlike the existing adaptive kernel SPH method, the fully variable smoothing lengths effects have been considered essentially in the scheme based on the adaptive symmetrical kernel estimation. Among the new equations, the evolution equation of density is derived in essence, it is implicitly coupled with variable smoothing length equation; the momentum equation and energy equation are derived from Springel’s fully conservative formulation SPH using the symmetrical kernel estimation instead of the scatter kernel estimation algorithm. Because the new SPH density evolution equation is implicitly coupled with variable smoothing length equation, an additional iteration process is employed necessary to solve the evolution equations of density and the variable smoothing lengths equation, and the SPH momentum equation and the energy equation is solved in nature, only the little cost is used in the iteration algorithm. The new equations and its algorithm are tested via two 1D shock-tube problems and a 2D Sedov problem, it is showed that the new algorithm corrects the variable smoothing lengths effect fairly well, especially in the 2D Sedov problem, the pressure peak is captured by the presented method more accurately than that of Springel’s scheme, and the accuracy of pressure at the center in Sedov problem is improved also. The new method can deal with the large density gradient and large smoothing length gradient problems well, such as large deformation and serious distortion problems in high velocity impact and blasting phenomenon.