Abstract
The upwind scheme is very important in the numerical approximation of some problems such as the convection dominated problem, the two-phase flow problem, and so on. For the fractional flow formulation of the two-phase flow problem, the Penalty Discontinuous Galerkin (PDG) methods combined with the upwind scheme are usually used to solve the phase pressure equation. In this case, unless the upwind scheme is taken into consideration in the velocity reconstruction, the local mass balance cannot hold exactly. In this paper, we present a scheme of velocity reconstruction in some H(div) spaces with considering the upwind scheme totally. Furthermore, the different ways to calculate the nonlinear coefficients may have distinct and significant effects, which have been investigated by some authors. We propose a new algorithm to obtain a more effective and stable approximation of the coefficients under the consideration of the upwind scheme.
Highlights
In the context of some fields, such as modeling and simulation of fluid flows in petroleum or groundwater reservoirs, the studies of processes of the simultaneous flow of two or more fluid phases within a porous medium are of great significance
How to cite this paper: Hou, J.Y., Yan, W.J. and Chen, J. (2015) Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem
A large number of methods, which are based on the finite difference (FD), the finite volume (FV) or the finite element (FE) methods, have been developed to deal with the two-phase flow problem
Summary
In the context of some fields, such as modeling and simulation of fluid flows in petroleum or groundwater reservoirs, the studies of processes of the simultaneous flow of two or more fluid phases within a porous medium are of great significance. (2015) Velocity Projection with Upwind Scheme Based on the Discontinuous Galerkin Methods for the Two Phase Flow Problem. In [2], an average total velocity was post processed by substituting the piecewise constants of pressure gradient and saturation gradient into the velocity-pressure expression directly Such reconstructed velocity, on some level, belongs to the lowest order Raviart-Thomas finite element space. We found that unless the upwind scheme and penalty terms which are used in the discretization of the two-phase flow problem are considered together into the velocity reconstruction, the error of the local mass conservation cannot reach a satisfactory level. We present a scheme of velocity reconstruction in some H(div) spaces [5] with considering the upwind scheme totally.
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