Stabilized and variationally consistent integrated meshfree formulation for advection-dominated problems

Abstract

Meshfree methods such as reproducing kernel (RK) approximation are suitable for modeling fluid flow problems because of their flexibility in controlling local smoothness and order of basis, as well as straightforward construction of higher-order gradients. However, Eulerian-described partial differential equations often suffer from numerical instability in the solution of Bubnov–Galerkin​ methods because of the non-self-adjoint advection terms. This is true for both mesh-based and meshfree methods. Although stabilized Petrov–Galerkin formulation, such as the variational multiscale method, has addressed this issue, a careless selection of the numerical quadrature can still result in variational inconsistency in the Galerkin weak form, which leads to a suboptimal convergence. Strong advection could also amplify the unstable modes from a reduced quadrature. This study provides a variationally consistent (VC) approach to correct the loss of Galerkin exactness in nodally integrated meshfree modeling for the advection diffusion equation. A gradient stabilization method is proposed to enhance the coercivity of the system. Several numerical examples are provided to verify the effectiveness and efficiency of the proposed approaches in modeling advection dominated problems.