Equivalent-PDE based stabilization of strong-form meshless methods applied to advection-dominated problems

摘要

In this paper a new criterion for stability of strong-form meshless discretizations of advection-dominated problems is developed. The criterion follows from the analysis of the equivalent PDE. It incorporates the influence of the positions of the nodes in the local subdomain, the influence of free parameters of local approximation, such as the weight function and the order of local approximation, and the time stepping formula used. We use the developed criterion in an adaptive upwinding scheme and test the developed scheme on a simple two-dimensional advection problem for regular and irregular node arrangement by using second-, third- and fourth-order accurate explicit diffuse approximate method. The proposed upwinding algorithm significantly improves the accuracy of the resulting solution in comparison with the existing approaches which do not consider the parameters of the local approximation and the irregular positioning of nodes, which is the usual setting for meshless methods.