Browsing by Author "Li, Jizhou"

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    Convergence of a high order method in time and space for the miscible displacement equations
    (EDP Sciences, 2015) Li, Jizhou; Riviere, Beatrice; Walkington, Noel
    A numerical method is formulated and analyzed for solving the miscible displacement problem under low regularity assumptions. The scheme employs discontinuous Galerkin time stepping with mixed and interior penalty discontinuous Galerkin finite elements in space. The numerical approximations of the pressure, velocity, and concentration converge to the weak solution as the mesh size and time step tend to zero. To pass to the limit a compactness theorem is developed which generalizes the Aubin-Lions theorem to accommodate discontinuous functions both in space and in time.
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    High order discontinuous Galerkin methods for simulating miscible displacement process in porous media with a focus on minimal regularity
    (2015-04-20) Li, Jizhou; Riviere, Beatrice M.; Symes, William; Hirasaki, George; Warburton, Timothy; Heinkenschloss, Matthias
    In my thesis, I formulate, analyze and implement high order discontinuous Galerkin methods for simulating miscible displacement in porous media. The analysis concerning the stability and convergence under the minimal regularity assumption is established to provide theoretical foundations for using discontinuous Galerkin discretization to solve miscible displacement problems. The numerical experiments demonstrate the robustness and accuracy of the proposed methods. The performance study for large scale simulations with highly heterogeneous porous media suggests strong scalability which indicates the efficiency of the numerical algorithm. The simulations performed using the algorithms for physically unstable flow show that higher order methods proposed in thesis are more suitable for simulating such phenomenon than the commonly used cell-center finite volume method.
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    Locally Mass-Conservative Method With Discontinuous Galerkin In Time For Solving Miscible Displacement Equations Under Low Regularity
    (2013-09-16) Li, Jizhou; Riviere, Beatrice M.; Heinkenschloss, Matthias; Symes, William W.; Warburton, Tim
    The miscible displacement equations provide the mathematical model for simulating the displacement of a mixture of oil and miscible fluid in underground reservoirs during the Enhance Oil Recovery(EOR) process. In this thesis, I propose a stable numerical scheme combining a mixed finite element method and space-time discontinuous Galerkin method for solving miscible displacement equations under low regularity assumption. Convergence of the discrete solution is investigated using a compactness theorem for functions that are discontinuous in space and time. Numerical experiments illustrate that the rate of convergence is improved by using a high order time stepping method. For petroleum engineers, it is essential to compute finely detailed fluid profiles in order to design efficient recovery procedure thereby increase production in the EOR process. The method I propose takes advantage of both high order time approximation and discontinuous Galerkin method in space and is capable of providing accurate numerical solutions to assist in increasing the production rate of the miscible displacement oil recovery process.
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    Strong Convergence of Discrete DG Solutions of the Heat Equation
    (2015-10) Girault, Vivette; Li, Jizhou; Riviere, Beatrice
    A convergence analysis to the weak solution is derived for interior penalty discontinuous Galerkin methods applied to the heat equation in two and three dimensions under general mixed boundary conditions. Strong convergence is established in the DG norm, as well as in the Lp norm, in space and in the L2 norm in time.