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

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