A Two-Grid Block-Centered Finite Difference Algorithm for Nonlinear Compressible Darcy–Forchheimer Model in Porous Media

Abstract

In this paper, a block-centered finite difference method is proposed to discretize the compressible Darcy–Forchheimer model which describes the high speed non-Darcy flow in porous media. The discretized nonlinear problem on the fine grid is solved by a two-grid algorithm in two steps: first solving a small nonlinear system on the coarse grid, and then solving a nonlinear problem on the fine grid. On the coarse grid, the coupled term of pressure and velocity is approximated by using the fewest number of node values to construct a nonlinear block-centered finite difference scheme. On the fine grid, the original nonlinear term is modified with a small parameter $$\varepsilon $$ to construct a linear block-centered finite difference scheme. Optimal order error estimates for pressure and velocity are obtained in discrete $$l^\infty (L^2)$$ and $$l^2(L^2)$$ norms, respectively. The two-grid block-centered finite difference scheme is proved to be unconditionally convergent without any time step restriction. Some numerical examples are given to testify the accuracy of the proposed method. The numbers of iterations are reported to illustrate the efficiency of the two-grid algorithm.