DMS Applied and Computational Mathematics Seminar

Speaker: Catalin Trenchea (University of Pittsburgh)  

Abstract: We propose and analyze a second-order partitioned time-stepping method for a two phase flow problem in porous media. The algorithm is based on a refactorization of Cauchy’s one-legged θ-method. The main part consists of the implicit backward Euler method, while part two uses a linear extrapolation. In the backward Euler step, the decoupled equations are solved iteratively. We prove that the iterations converge linearly to the solution of the coupled problem, under some conditions on the data. When θ=1/2, the algorithm is equivalent to the symplectic midpoint method. Similar to the continuous case, we also prove a discrete Helmholtz free energy balance, without numerical dissipation. We compare this midpoint method with the classic backward Euler method, and two implicit-explicit time-lagging schemes. The midpoint method outperforms the other schemes in terms of rates of convergence, long-time behaviour and energy approximation, for small and large values of the time step.