Browsing by Author "Yang, Xin"

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    A coupled finite volume and discontinuous Galerkin method for convection-diffusion problems
    (2012) Yang, Xin; Riviere, Beatrice M.
    This work formulates and analyzes a new coupled finite volume (FV) and discontinuous Galerkin (DG) method for convection-diffusion problems. DG methods, though costly, have proved to be accurate for solving convection-diffusion problems and capable of handling discontinuous and tensor coefficients. FV methods have proved to be very efficient but they are only of first order accurate and they become ineffective for tensor coefficient problems. The coupled method takes advantage of both the accuracy of DG methods in the regions containing heterogeneous coefficients and the efficiency of FV methods in other regions. Numerical results demonstrate that this coupled method is able to resolve complicated coefficient problems with a decreased computational cost compared to DG methods. This work can be applied to problems such as the transport of contaminant underground, the CO 2 sequestration and the transport of cells in the body.
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    Simulation of CO2 sequestration in saline aquifers using discontinuous Galerkin method
    (2014-08-01) Yang, Xin; Riviere, Beatrice M.; Symes, William W; Warburton, Timothy; Verduzco, Rafael
    Carbon dioxide disposal into deep aquifer has been an important venue to trap excess gas emission which causes global warming. In the CO2 sequestration process, CO2 is captured from the point source and injected into the saline aquifer deep enough where CO2 reaches the supercritical state and it has a very high density compared to gaseous state. This process is described by the two-phase two-component model, which involves two nonlinear time dependent advection-diffusion equations. The difficulty lies in the injection phase when the advection terms highly dominate over the diffusion terms. Discontinuous Galerkin (DG) methods, which are famous for the properties of high order accuracy, locality and locally mass conservation, have proved to be promising for advection dominated transport equations. I develop a new fully implicit fully coupled DG method called “partial upwind” method, to discretize the equations. For time discretization, it uses the backward Euler method to allow large time steps. For space discretization, it uses the usual interior penalty DG discretization for the elliptic terms and the upwind for part of the advection terms. The other part of the advection terms are handled specially for stabilization purpose. Numerical simulations show that the new DG method works well for the CO2 sequestration problems in homogenous porous media and has shown great potential in heterogenous porous media. I also compare the new method with the primal interior penalty DG method and show that the new method is superior to the usual DG method for some subsurface fluid flow problems. Though DG methods perform well for the CO2 sequestration problem, they are indeed more costly than traditional numerical methods. The first order finite volume (FV) method, on the other hand, is very efficient. A new coupled finite volume and discontinuous Galerkin method, which uses the accuracy of DG methods on some parts of the domain and the efficiency of FV methods everywhere else to reduce the computational cost, is also studied for the time-dependent advection-diffusion equations. Theoretical and numerical results show that the new coupled method converges and can be both accurate and efficient at the same time for some typical examples. We want to apply the new coupled method to the CO2 sequestration problems in the future.
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    Simulation of CO2 Sequestration in Saline Aquifers Using Discontinuous Galerkin Method
    (2014-08) Yang, Xin
    Carbon dioxide disposal into deep aquifer has been an important venue to trap excess gas emission which causes global warming. In the CO2 sequestration process, CO2 is captured from the point source and injected into the saline aquifer deep enough where CO2 reaches the supercritical state. This process is described by the two-phase two-component model, which involves two nonlinear time dependent advection-diffusion equations. The difficulty lies in the injection phase when the advection terms highly dominate over the diffusion terms. Discontinuous Galerkin (DG) methods, which are famous for the properties of high order accuracy, locality and locally mass conservation, have proved to be promising for advection dominated transport equations. I develop a new fully-implicit fully-coupled method with the so called "partial upwind" DG method for space discretization to solve the equations. For time discretization, it uses the backward Euler method to allow large time steps. For space discretization, it uses the usual interior penalty DG discretization for the elliptic terms and the upwind for part of the advection terms. The other part of the advection terms are handled specially for stabilization purpose. Numerical simulations show that the new DG method works well for the CO2 sequestration problems in homogenous porous media and shows great potential in heterogeneous porous media. I also compare the new method with the primal interior penalty DG method, and show that the new method is superior to the usual DG method for some subsurface fluid flow problems. Though DG methods perform well for the CO2 sequestration problem, they are indeed more costly than traditional numerical methods. The first order finite volume (FV) method, on the other hand, is very efficient. A new coupled finite volume and discontinuous Galerkin method, which uses the accuracy of DG methods on some parts of the domain and the efficiency of FV methods everywhere else to reduce the computational cost, is also studied for the time-dependent advection-diffusion equations. Theoretical and numerical results show that the new coupled method converges and can be both accurate and efficient at the same time for some typical examples. We want to apply the new coupled method to the CO2 sequestration problems in the future.