Generalized Finite Difference Method for Numerical Solutions of Density-driven Groundwater Flows

Abstract

A combination of the generalized finite difference method (GFDM), the implicit Euler method and the Newton-Raphson method is proposed to efficiently and accurately analyze the density-driven groundwater flows. In groundwater hydraulics, the problems of density-driven groundwater flows are usually difficult to be solved, since the mathematical descriptions are a system of time- and space-dependent nonlinear partial differential equations. In the proposed numerical scheme, the GFDM and the implicit Euler method were adopted for spatial and temporal discretizations of governing equations. The GFDM is a newly-developed meshless method and is truly free from time-consuming mesh generation and numerical quadrature. Based on the concept of star in the GFDM and the moving-least-squares method, the derivatives with respect to space coordinates at every node are expressed by linear combinations of nearby function values with different weighting coefficients. After discretizations of the GFDM and the implicit Euler method, a system of nonlinear algebraic equations at every time step is yielded and then can be efficiently solved by the Newton-Raphson method. Two numerical tests including the Henry and the Elder problems were adopted to verify the accuracy and the stability of the proposed meshless numerical scheme. Besides, the numerical results were compared with other numerical and semi-analytical solutions.