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The error of smoothed particle hydrodynamics (SPH) using kernel forparticle-based approximation mainly comes from smoothing and integrationerrors. The choice of kernels has a significant impact on the numericalaccuracy, stability and computational efficiency. At present, the most popularkernels such as B-spline, truncated Gaussian (for compact support), Wendlandkernels have 2nd-order smoothing error and Wendland kernel becomes mainstreamin SPH community as its stability and accuracy. Due to the fact that theparticle distribution after relaxation can achieve fast convergence ofintegration error respected to support radius, it is logical to choose kernelswith higher-order smoothing error to improve the numerical accuracy. In thispaper, the error of 4th-order Laguerre-Wendland kernel proposed by Litvinov etal. \cite{litvinov2015towards} is revisited and another 4th-order truncatedLaguerre-Gauss kernel is further analyzed and considered to replace the widelyused Wendland kernel. The proposed kernel has following three properties: Oneis that it avoids the pair-instability problem during the relaxation process,unlike the original truncated Gaussian kernel, and achieves much lessrelaxation residue than Wendland and Laguerre-Wendland kernels; One is thetruncated compact support size is the same as the non-truncated compact supportof Wendland kernel, which leads to both kernels' computational efficiency atthe same level; Another is that the truncation error of this kernel is muchless than that of Wendland kernel. Furthermore, a comprehensive set of $2D$ and$3D$ benchmark cases on Eulerian SPH for fluid dynamics and total LagrangianSPH for solid dynamics validate the considerably improved numerical accuracy byusing truncated Laguerre-Gauss kernel without introducing extra computationaleffort.

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