An iteration-free semi-Lagrangian meshless method for Burgers' equations

It has been verified in our previous study, the semi-Lagrangian method combined with the element free Galerkin method performs very well in solving the convection-dominated problems. The meshless feature facilitates the backtracking process arising in the semi-Lagrangian method because when interpolating the function values to a point it is needless to expend much effort in determining which grid cell contains the point. However, the moving least squares approximation used in the element free Galerkin method is somewhat expensive. Besides, if there are many iterations during the backtracking process, efficiency of the method will be a bit low, especially in case of tracking the quadrature nodes. For this reason, this paper aims to present an iteration-free backtracking method to get rid of the iterations and hence improve the computational efficiency. The new method is based on Taylor expansion, the idea of which is simple and very easy to implement. To demonstrate the high efficiency and precision of the new method, we compare it with three other backtracking methods by solving several inviscid as well as viscous Burgers' equations. Results show that this method greatly improves the computational efficiency of the backtracking method while maintaining high accuracy.

Automated summary

Based on our previous study, the combination of the semi-Lagrangian method and the element free Galerkin method has been proven to be effective in solving convection-dominated problems. The meshless feature simplifies the backtracking process in the semi-Lagrangian method. However, the moving least squares approximation in the element free Galerkin method is somewhat costly. To address this issue, this paper presents an iteration-free backtracking method based on Taylor expansion, which greatly improves computational efficiency without sacrificing accuracy when compared to other backtracking methods.