A new Runge–Kutta–Chebyshev Galerkin-characteristic finite element method for advection–dispersion problems in anisotropic porous media

We propose a new approach that combines the modified method of characteristics with a unified finite element discretization for the numerical solution of a class of coupled Darcy-advection–dispersion problems in anisotropic porous media. The proposed method benefits from advantages of the method of characteristics in its ability to handle the nonlinear convective terms, while taking advantage of a unified formulation that allows the use of equal-order finite element approximations along with the L2-projection method for all solutions in the problem. In the proposed Galerkin-characteristic finite element framework, the standard Courant–Friedrichs–Lewy condition is relaxed, and time truncation errors are reduced since no stability criterion restricts the choice of time step. For time integration, we use a Runge–Kutta scheme with the Chebyshev polynomials (RKC). The RKC method has an extensive stability field and it is explicit and second-order accurate. In order to assess the quality of the proposed approach, we perform numerical experiments for several categories of test cases. The first category concerns accuracy test examples with known analytical solutions, while the second category focuses on a benchmark problem of miscible flow in a heterogeneous porous medium. The numerical results obtained show high performance and demonstrate the ability of the method to capture dispersion effects in porous media problems.