Numerical Analysis of a Discontinuous Galerkin Method for Multiphase Flow in Porous Media
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Modeling the flow of liquid, aqueous, and vapor phases through porous media is challenging from the computational and theoretical point of view, as it requires solving nonlinear coupled partial differential equations. In this talk, we propose a second-order accurate and energy-stable time discretization method for the three-phase flow problem in porous media. We prove the convergence of the subiterations to resolve the nonlinearity, and show that the time-stepping method mimics the energy balance relation that the continuous problem satisfies. Our spatial discretization uses an interior penalty discontinuous Galerkin method, for which we establish the well-posedness of the discrete problem and provide error estimates under certain conditions on the data. We validate our method through numerical simulations, which show that our approach achieves the expected theoretical convergence rates.