An Adaptive Explicit Runge-Kutta Finite Element Semi-Lagrangian Method for Anisotropic Transport-Dispersion in Porous Media

We propose a new adaptive explicit Runge-Kutta finite element semi-Lagrangian method for the numerical solution of a time-dependent transport-dispersion problem in an anisotropic porous medium. The governing equations consist of the Darcy equations for the pressure and flow field coupled to a transport-dispersion equation for the solute concentration. The proposed method is a combination of the modified characteristic method used for the integration of the total derivative in time along with an unified finite element discretization for all the solution variables of the problem under consideration. The L2 -projection is also implemented for the evaluation of the numerical solutions at the departure points of each integration point in the host element. To further increase the accuracy and efficiency in the proposed method, we implement local enrichments based on adaptive quadrature rules using a residual-based error estimator. Adaptive multi-level enrichment techniques are also used in the present study. The aim is to update the quadrature points as required without adapting the mesh during the time integration procedure. This ensures that the linear systems associated with the enriched finite element semi-Lagrangian method maintain the same structure and size throughout the adaptation process. The new approach preserves the expanded stability property of the Runge-Kutta-Chebyshev method for solving ordinary differential equations, and it only requires a single projection per time step. In order to evaluate the accuracy of the proposed approach, we present the numerical solution of two transport-dispersion problems in anisotropic media and the obtained results are compared to those obtained using the conventional Galerkin finite element method. The numerical results show that the proposed approach allows time-dependent transport-dispersion problems to be solved efficiently and with an appropriate level of accuracy.