A Hybrid Interpolating Meshless Method for 3D Advection–Diffusion Problems

A hybrid interpolating meshless (HIM) method is established for dealing with three-dimensional (3D) advection–diffusion equations. To improve computational efficiency, a 3D equation is changed into correlative two-dimensional (2D) equations. The improved interpolating moving least-squares (IIMLS) method is applied in 2D subdomains to obtain the required approximation function with interpolation property. The finite difference method (FDM) is utilized in time domain and the splitting direction. Setting diagonal elements to one in the coefficient matrix is chosen to directly impose Dirichlet boundary conditions. Using the HIM method, difficulties created by the singularity of the weight functions, such as truncation error and calculation inconvenience, are overcome. To prove the advantages of the new method, some advection–diffusion equations are selected and solved by HIM, dimension splitting element-free Galerkin (DSEFG), and improved element-free Galerkin (IEFG) methods. Comparing and analyzing the calculation results of the three methods, it can be shown that the HIM method effectively improves computation speed and precision. In addition, the effectiveness of the HIM method in the nonlinear problem is verified by solving a 3D Richards’ equation.