An upwind moving least squares approximation to solve convection-dominated problems: An application in mixed discrete least squares meshfree method

Moving Least Squares (MLS), as a series representation type of approximation, is broadly used in a wide array of meshfree methods. However, using the existing standard form of the MLS causes an unphysical oscillation for the meshfree methods encountering the convection-dominated partial differential equations (PDEs). In this study, several approaches are investigated to enhance the MLS approximation for solving convection-dominated problems. A novel upwind version of MLS approximation called shifted upward MLS (SU-MLS), is presented, which is based on strengthening the effect of upwind nodal points in approximation by wisely adjusting the weight function. Regarding the presented theoretical/numerical investigation, the proposed SU-MLS approximation can yield monotone solutions encountering the convection-dominated PDEs, unlike the existing standard form of MLS. The suggested SU-MLS approximation is, then, utilized in the mixed discrete least squares meshfree (MDLSM) method, rather than the standard MLS which is conventionally used in existing MDLSM. The novel method, which uses SU-MLS, is named upwind MDLSM (UMDLSM). Several numerical examples are investigated, and the results are compared to existing MDLSM. The obtained results indicate that the suggested UMDLSM is remarkably more accurate than the existing MDLSM in convection-dominated PDEs. Furthermore, while the existing MDLSM dramatically suffers from spurious oscillations (wiggling) when the Peclet number is high, the presented UMDLSM can yield monotone and accurate solutions.