A Viscosity-Splitting Method for the Navier-Stokes/ Darcy Problem

Abstract

In this report, we give a viscosity splitting method for the Navier-Stokes/Darcy problem. In this method, the Navier-Stokes/Darcy equation is solved in three steps. In the first step, an explicit/ implicit formulation is used to solve the nonlinear problem. We introduce an artificial diffusion term $\theta\Delta\u00a0 \mathbf{u}$ in our scheme whose purpose is to enlarge the time stepping and enhance numerical stability, especially for small viscosity parameter $\ u$, by choosing suitable parameter $\theta$. In the second step, we solve the Stokes equation for velocity and pressure. In the third step, we solve the Darcy equation for the piezometric head in the porous media domain. We use the numerical solutions at last time level to give the interface condition to decouple the Navier-Stokes equation and the Darcy's equation. The stability analysis, under some condition $\Delta\leq k_0$, $k_0>0$, is given. The error estimates prove our method has an optimal convergence rates. Finally, some numerical results are presented to show the performance of our algorithm.