A weighted–upwind generalized finite difference (WU–GFD) scheme with high–order accuracy for solving convection–dominated problems

Abstract

In this study, a novel upwind numerical scheme based on meshless framework was proposed to solve convection–dominated problems in one– and two–dimensions. The proposed numerical scheme improved the star structure in the generalized finite difference method (GFDM) by incorporating a weighted–upwind approach. The localization and meshless characteristics of the GFDM make it easy to adapt to the improvement of the upwind star, and the derivatives can be discretized by collecting the supporting nodes on the upwind direction. The weighted–upwind approach allows the proposed numerical scheme to adjust between fully upwind scheme and weighted upwind scheme by expressing. Therefore, the numerical discretization of the convective term can be as a linear combination of derivative approximations using neighboring nodes both in the upwind and original stars. In contrast to traditional upwind methods offering only first-order accuracy, this numerical approach achieves a higher–order level of accuracy. Two numerical examples validate the proposed meshless upwind scheme, highlighting its accuracy through convergence rate tests. The study further fills the gap in applying GFDM–type schemes to convection–dominated problems, emphasizing the need for enhanced meshless approaches in such applications.