Assessment of Localization Strategies in a Radial Basis Function Meshless Method to Solve Two-Dimensional Convection-Diffusion Problems

Two-dimensional Poisson and convection-diffusion problems are solved by using three localization schemes implemented in the context of a Radial Basis Function (RBFs) collocation method. The first scheme uses the traditional RBF superpositions to approximate the problem variable in a defined stencil. The second scheme is the Partition of Unity strategy and it is used to obtain a representation of governing equations as a linear combination of RBFs local superpositions evaluated at neighbouring stencils. Weight functions are designed to capture the convection term effect on the solution. In the third scheme, an upwind strategy is included in the Partition of Unity scheme for solving the convection-diffusion problem by moving and deforming stencils based on velocity. For all schemes, stencils in the form of crosses, circles, and squares are considered, and Root mean square (RMS) is obtained as a function of shape parameter, nodal distribution size and stencil size. In the case of Poisson problems, the use of Partition of unity in circular configuration with no more than 37 nodes per stencil is recommended as far as a c suitable range is obtained for each nodal distribution employed to avoid ill-conditioning issues.