Mixed Finite Element-Second Order Upwind Fractional Step Difference Scheme of Darcy–Forchheimer Miscible Displacement and Its Numerical Analysis

Abstract

The incompressible miscible displacement of three-dimensional Darcy–Forchheimer flow is discussed in this paper, and the mathematical model is formulated by two partial differential equations, a Darcy–Forchheimer flow equation for the pressure and a convection-diffusion equation for the concentration. The pressure plays an important role and determines the Darcy–Forchheimer velocity. A conservative mixed finite element is used to approximate the pressure and the velocity, and the computational accuracy of Darcy–Forchheimer velocity is improved by one order. A second-order upwind fractional step difference scheme is adopted to obtain the concentration, and numerical oscillation and dispersion are eliminated. Computational work is reduced greatly by decomposing the whole computation into successive one-dimensional subcomputations, where the speedup solvers are used. Applying energy-norm method, operator decomposition, speedup algorithm and induction hypotheses, we derive an optimal second-order estimates in $$L^2$$ norm. Numerical experiments are given to show the theoretical accuracy and the applicability for solving actual problems.