A fourth-order kernel for improving numerical accuracy and stability in Eulerian SPH for fluids and total Lagrangian SPH for solids

The error of smoothed particle hydrodynamics (SPH) using a kernel for particle-based approximation mainly arises from smoothing and integration errors. The choice of the kernel significantly impacts numerical accuracy, stability and computational efficiency. Currently, the most popular kernels, such as B-spline, truncated Gaussian (for compact support), and Wendland kernels, have 2nd-order smoothing errors, and the Wendland kernel has become mainstream in the SPH community due to its stability and accuracy. Since the particle distribution after relaxation can achieve fast convergence of integration error with respect to support radius, it is logical to choose kernels with higher-order smoothing error to improve the numerical accuracy. In this paper, the error of the 4th-order Laguerre-Wendland kernel proposed by Litvinov et al. [1] is revisited, and another 4th-order truncated Laguerre-Gauss kernel is further analyzed and considered to replace the widely used Wendland kernel. The proposed kernel has the following three properties: One is that it avoids the pair-instability problem during the relaxation process, unlike the original truncated Gaussian kernel, and achieves much less relaxation residue than the Wendland and Laguerre-Wendland kernels; One is the truncated compact support size is the same as the non-truncated one of the Wendland kernel, which leads to both kernels' comparably computational efficiency; Another is that the truncation error of this kernel is much less than that of the Wendland kernel. Furthermore, a comprehensive set of 2D and 3D benchmark cases on Eulerian SPH for fluid dynamics and total Lagrangian SPH for solid dynamics validate the considerably improved numerical accuracy of the truncated Laguerre-Gauss kernel without introducing extra computational effort.