Numerical solutions of the incompressible miscible displacement equations in heterogeneous media

•We solve the miscible displacement problem.•We combine mixed finite elements with discontinuous Galerkin in space.•We use Runge–Kutta methods in time.•We model the flow on realistic heterogeneous media.

This paper presents a numerical method based on mixed finite element, discontinuous Galerkin methods in space and high order Runge–Kutta method in time for solving the miscible displacement problem. No slope limiters are needed. The proposed method exhibits high order of convergence in space and time when comparing with analytical solutions. The simulation shows robustness of the method for heterogeneous media with highly varying permeabilities.