A localized RBF-MLPG method for numerical study of heat and mass transfer equations in elliptic fins

Abstract

The aim of this paper is to propose a local meshless method for the numerical solutions of heat and mass transfer equations in elliptic fins. The domain and boundary are discretized with arbitrary positioned nodes. Then, around each node a small number of neighboring nodes, called stencil, is considered. By collocating the unknown solution and enforcing the PDE governing operator on each stencil, a small system is obtained. Enforcement of the PDE governing operator on each stencil is done via its local weak formulation on a subdomain contained in the stencil.Using local weak formulation increases the stability of the method and makes it more efficient than the existing Local RBF methods. In addition, the boundary conditions are directly imposed via stencils which contain boundary points. Using the concept of generalized differentiations, an iterative approach based on Newton method is proposed to deal with the nonlinear term. Abenchmark problem with known analytical solution is used to investigate the accuracy of the proposed method and its sensitivity to different parameters. It is observed that the method enjoys exponential convergence and in comparison with existing local methods, the presented method is more stable and accurate. After investigating the validation of the method, the method is successfully used for solving the governing differential equations of the 2-D temperature distribution under the dry and fully wet conditions and the numerical results are obtained and presented on two different computational domains.