A Local Meshless Method for Steady State Convection Dominated Flows

Abstract

A stable localized meshless method (SLMM) is proposed for convection dominated PDEs exhibiting boundary layer. Some cases of continuous and discontinuous boundary data as well as continuous and discontinuous source function with constant and variable convection coefficients are considered. In this approach, the localized meshless method is implemented on specialized sub-domains embedded with flow direction of underlying fluid. The proposed method is based on flow featured overlapping sub-domains, called stencils. The concept of flow direction is used to construct good quality stencils having the ability to capture flow features, such as boundary layer, accurately. Numerical experiments are presented to compare the proposed method with the finite-difference method on special grid (FDSG), the standard finite-element method, hybridized SUPG method, hybridized upwind method, residual-free bubbles (RFB) method and other meshless methods. Numerical results confirm that the new approach is accurate and efficient for solving a wide class of one-, two-, and three-dimensional convection-dominated PDEs. In some cases, performance of the SLMM is comparable and sometimes better than the mesh-based finite-element and finite-difference methods.

Local meshless methods for second order elliptic interface problems with sharp corners

“…Another benefit of the localized methods is the flexibility of the stencil selection and the nodal point selection to improve accuracy of the method. For instance, nodes can be used in the local support domain on the upwind side to obtain a physically viable solution in the case of some boundary layer problems and convection-dominated PDEs [23,[29][30][31].…”