A semi-Lagrangian meshfree Galerkin method for convection-dominated partial differential equations

Abstract

A new numerical method, which is based on the coupling between semi-Lagrangian (SL) method and element free Galerkin (EFG) method, is developed for convection–diffusion partial differential equations with dominated convection terms. It is the first time that the SL method combined with the EFG method to solve the transient convection–diffusion equations. This combination has been more effective than expected. Compared with the existing SL methods, the proposed method shows several new advantages. Firstly, it takes full advantages of meshfree methods, therefore, the implementation difficulties of the SL method in backward tracing and departure point interpolation are greatly reduced. As interpolating the function values to a departure point, complex searching algorithm is not needed to determine which grid cell contains the departure point, and higher order interpolation is not essential to ensure the global accuracy. Meanwhile, in addition to having good stability and allowing the use of large time-step sizes, the new method realizes the continuous transition from convection–diffusion equation to pure convection equation and provides a uniform numerical format for these two kinds of equations. A problem with available analytical solution is solved to analyze the convergence behavior of the proposed method. It is demonstrated that the method can achieve the optimal convergence rate. Subsequently, the method is applied to solve the 1D and 2D Burgers’ equations. The results show that the proposed method can indeed obtain surprisingly good results for the Burgers’ equations in the very challenging convection-dominated and pure convection cases. These results are better than almost all the ones we have seen in literature. The good performance of this method in solving the challenging Burgers’ equations makes us believe that it will be a very promising method for solving convection-dominated problems.