In this paper we present a second-order accurate adaptive algorithm for solving multiphase, incompressible flows in porous media. We assume a multiphase form of Darcy's law with relative permeabilities given as a function of the phase saturation. The remaining equations express conservation of mass for the fluid constituents. In this setting the total velocity, defined to be the sum of the phase velocities, is divergence-free. The basic integration method is based on a total-velocity splitting approach in which we solve a second-order elliptic pressure equation to obtain a total velocity. This total velocity is then used to recast component conservation equations as nonlinear hyperbolic equations. Our approach to adaptive refinement uses a nested hierarchy of logically rectangular grids with simultaneous refinement of the grids in both space and time. The integration algorithm on the grid hierarchy is a recursive procedure in which coarse grids are advanced in time, fine grids are advanced multiple steps to reach the same time as the coarse grids and the data at different levels are then synchronized. The single grid algorithm is described briefly, but the emphasis here is on the time-stepping procedure for the adaptive hierarchy. Numerical examples are presented to demonstrate the algorithm's accuracy and convergence properties and to illustrate the behavior of the method.

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# Your search: "author:"Pau, George Shu Heng""

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Ballistic transport simulation in nanodevices, which involves self-consistently solving a coupled Schrodinger-Poisson system of equations, is usually computationally intensive. Here, we propose coupling the reduced basis method with the subband decomposition method to improve the overall efficiency of the simulation. By exploiting a posteriori error estimation procedure and greedy sampling algorithm, we are able to design an algorithm where the computational cost is reduced significantly. In addition, the computational cost only grows marginally with the number of grid points in the confined direction.