An analysis of the convection–diffusion problems using meshless and meshbased methods

The numerical solution of the convection–diffusion equation represents a very important issue in many numerical methods that need some artificial methods to obtain stable and accurate solutions. In this article, a meshless method based on the local Petrov–Galerkin method is applied to solve this equation. The essential boundary condition is enforced by the transformation method, and the MLS method is used for the interpolation schemes. The streamline upwind Petrov–Galerkin (SUPG) scheme is developed to employ on the present meshless method to overcome the influence of false diffusion. In order to validate the stability and accuracy of the present method, the model is used to solve two different cases and the results of the present method are compared with the results of the upwind scheme of the MLPG method and the high order upwind scheme (QUICK) of the finite volume method. The computational results show that fairly accurate solutions can be obtained for high Peclet number and the SUPG scheme can very well eliminate the influence of false diffusion.

► MLPG method is applied to solve convection-dominated problems.
► SUPG scheme is constructed in the MLPG method.
► High order scheme of FVM is applied to compare with the MLPG/SUPG.