Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media

The purposes of this work are to study the $L^{2}$-stability a Navier-Stokes type model for non-stationary flow in porous media proposed by Hsu and Cheng 1989 develop Lagrange-Galerkin scheme with Adams-Bashforth method solve that numerically.The stability estimate is obtained thanks presence nonlinear drag force term which corresponds Forchheimer term. We derive extending idea characteristics overcome difficulty comes from non-homogeneous porosity. Numerical experiments conducted investigate experimental order convergence scheme. For both simple complex designs porosities, our numerical simulations exhibit natural profiles well describe media.