A stabilized semi-Lagrangian finite element method for natural convection in Darcy flows

Abstract

We present an accurate semi-Lagrangian finite element method for the numerical solution of groundwater flow problems in porous media with natural convection. The mathematical model consists of the Darcy problem for the flow velocity and pressure subject to the Boussinesq approximation of low density variations coupled to a convection–diffusion equation for the concentration. The main idea is to combine the semi-Lagrangian method for time integration with finite element method for space discretization, so that the standard Courant–Friedrichs–Lewy condition is relaxed and the time truncation errors are reduced in the diffusion part of the governing equations. We also use a local L²-projection stabilization technique in order to improve the accuracy of the presented method. Numerical simulations are carried out for a test example of a natural convection in an aquifer system with natural boundaries. The obtained results demonstrate the ability of the proposed semi-Lagrangian finite element method to offer efficient and accurate simulations for natural convection in Darcy flows.