Generalized finite difference method for solving the double-diffusive natural convection in fluid-saturated porous media

In this paper, the generalized finite difference method (GFDM) combined with the Newton-Raphson method is proposed to accurately and efficiently simulate the steady-state double-diffusive natural convection in parallelogrammic enclosures filled with fluid-saturated porous media. The natural convection in fluid-saturated porous media, which is interesting in regard to the heat-transferring range, involves different physical compositions to affect the fluid flow. For the mathematical formulations of the natural convention, the governing equations are a system of highly-nonlinear partial differential equations, so the approximate solutions for the natural convention mainly depend on a suitable numerical scheme. In this study, the GFDM, a newly-developed meshless method, is adopted for the spatial discretization of the non-linear governing equations, since it can avoid setting up the mesh in the computational domain and implementing the numerical quadrature. The localization of the GFDM will result in a sparse system, while the derivatives at each node can be expressed as linear combinations of nearby function values with different weighting coefficients. After a system of nonlinear algebraic equations is yielded by the spatial discretization of the GFDM, the two-steps Newton-Raphson method is adopted to efficiently solve this resultant sparse system owing to the localization of the GFDM. Three numerical examples are presented to demonstrate the applicability and stability of the proposed meshless numerical scheme. Besides, the numerical results are compared with other solutions to show the accuracy of the proposed method.