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 L 2 -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.

A Cell-Centered Semi-Lagrangian Finite Volume Method for Solving Two-Dimensional Coupled Burgers’ Equations

“…The time derivative and the advection terms are associated in a directional derivative manner along the characteristics leading to a characteristic time-stepping procedure. The time truncation errors are greatly reduced due to the Lagrangian treatment, see, for instance, [20][21][22][23]. In addition, the semi-Lagrangian method offers the possibility of using time steps that go beyond those allowed by the CFL stability conditions in Eulerian finite element methods for convectiondominated flows.…”

Analysis of a Galerkin-characteristic finite element method for convection-diffusion problems in porous media

“…Moreover, it offers the possibility of using time steps that exceed those allowed by the stability CFL condition for the conventional Eulerian methods. The Galerkin-characteristic methods have been subject of investigations in many references including [6,19,20,21,36,37,38,42,43,44,46]. In [19], the semi-Lagrangian method combined with a finite difference discretization has been studied and applied to convection-diffusion problems.…”

“…In [23,25], the method has proved to be stable and accurate when used to study the thermal incompressible Navier-Stokes equations. Semi-Lagrangian methods have also been investigated for the simulation of natural and mixed convection flows in [21,43], for tidal flows in [24], and for moving thermal fronts in porous media in [42]. A first-order Galerkin-characteristic method combined with the finite element discretization has been analyzed for the Navier-Stokes equations in [38].…”