An Efficient Second-Order Algorithm Upon MAC Scheme for Nonlinear Incompressible Darcy–Brinkman–Forchheimer Model

In this paper, the Marker and Cell scheme based on a two-grid algorithm is proposed for the two-dimensional incompressible Darcy–Brinkman–Forchheimer equations in porous media. The motivation of the two-grid Marker and Cell algorithm is figuring out a nonlinear equation on a coarse grid with mesh size H and a linear equation on a fine grid with mesh size h. A small positive parameter \(\varepsilon \) $\epsilon$ is introduced. By using it, the non-differentiable nonlinear term can be transformed into the term which is twice continuously differentiable. The error estimates of the velocity and pressure in the \(L^2\) $L^2$ norms are obtained, which show \(O(\varepsilon +H^4+h^2)\) $O(\epsilon +H^4+h^2)$ . Second-order accuracy for some terms of velocity in the \(H^1\) $H^1$ norms is also obtained. Several numerical experiments are provided to confirm the availability of this efficient second-order algorithm. Behavior of the fluid flow with different Brinkman number is considered.