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A meshless local Petrov–Galerkin (MLPG) approach based on the streamline upwind (SU) idea and the variational multiscale (VMS) concept, called as VMS-SUMLPG method, is herein proposed to solve the convection-dominated problems. In the present VMS-SUMLPG method, the streamline upwind is constructed in the test function to solve the non-self-adjoint matrix. Meanwhile, the VMS concept as a stability term is adopted to alleviate the numerical instability such as spurious oscillations, overshoots, and undershoots. Its numerical accuracy and stability are validated by comparing with the streamline upwind Petrov–Galerkin (SUMLPG) method and the finite volume method with high-order difference schemes for two classical convection-dominated problems at the Peclet number ranging from 106to 108. It is shown that the numerical solutions of the present VMS-SUMLPG method are accuracy, smoothness, and stability. [ABSTRACT FROM PUBLISHER]

It is well known that convection-diffusion equations suffer from the most difficult problems to gain a stable and accurate solution. Sometimes, the convection terms may cause oscillatory behavior of solutions. In this article, the streamline upwind Petrov-Galerkin (SUPG) scheme is applied to eliminate overshoots and undershoots produced by the convection term in the meshless local Petrov-Galerkin (MLPG) method. The accuracy and stability of the present method are discussed using two test cases. The results of the present method are compared with results of other upwind schemes and the finite volume method (FVM) using a high-order upwind scheme. Our analysis shows that the present method, with its simple implementation, can give excellent results for convection-dominated problems. [ABSTRACT FROM PUBLISHER]