In this paper, we present a hybrid method, which consists of a mixed-hybrid finite element method and a penalty discontinuous Galerkin method, for the approximation of a fractional flow formulation of a two-phase flow problem in heterogeneous media with discontinuous capillary pressure. The fractional flow formulation is comprised of a wetting phase pressure equation and a wetting phase saturation equation which are coupled through a total velocity and the saturation affected coefficients. For the wetting phase pressure equation, the continuous mixedhybrid finite element method space can be utilized due to a fundamental property that the wetting phase pressure is continuous. While it can reduce the computational cost by using less degrees of freedom and avoiding the post-processing of velocity reconstruction, this method can also keep several good properties of the discontinuous Galerkin method, which are important to the fractional flow formulation, such as the local mass balance, continuous normal flux and capability of handling the discontinuous capillary pressure. For the wetting phase saturation equation, the penalty discontinuous Galerkin method is utilized due to its capability of handling the discontinuous jump of the wetting phase saturation. Furthermore, an adaptive algorithm for the hybrid method together with the centroidal Voronoi Delaunay triangulation technique is proposed. Five numerical examples are presented to illustrate the features of proposed numerical method, such as the optimal convergence order, the accurate and efficient velocity approximation, and the applicability to the simulation of water flooding in oil field and the oil-trapping or barrier effect phenomena.Keywords. Brezzi-Douglas-Marini, Raviart-Thomas, mixed-hybrid, continuity of wetting phase pressure, penalty discontinuous Galerkin, discontinuous nonlinear interface condition, centroidal Voronoi Delaunay triangulation § 1 IntroductionIn the article we focus on the numerical methods for the model of fractional flow formulation of a two-phase flow problem. The fractional flow formulation has two equations, the pressure equation and the saturation equation which are coupled through the total velocity and the saturation affected coefficients. Depending on different combination of the basic equations of the two-phase flow
We study a discrete-time model with diffusion that describe the dynamics of viral infections by using nonstandard finite difference (NSFD) scheme. The original model we considered was a viral infection model with cellular infection and general nonlinear incidence. We analyze thoroughly the dynamical properties of both discrete and original continuous models and show that the discrete system is dynamically consistent with the original continuous model, including positivity and boundedness of solutions, equilibria, and their global properties. The results imply that the NSFD scheme can efficiently preserve the global dynamics properties of the corresponding continuous model. Some numerical simulations are carried out to validate the theoretical results.
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