Abstract
As the number of processors on supercomputers has increased dramatically, there is a growing interest in developing scalable algorithms with a high degree of parallelism for extreme-scale reservoir simulation. However, traditional simulators and algorithms for such nonlinear problems are usually based on the family of time-marching methods, where parallelization is restricted to the spatial dimension only. In this paper, we propose a family of parallel-in-time (PinT) reservoir algorithms for solving multiphase flow problems in porous media to fully exploit the parallelism of supercomputers, in which the design of efficient nonlinear and linear space-time algorithms plays an essential role. More precisely, we first introduce a space-time mixed finite element method for the fully-implicit discretization. The nonlinear system arising at each parallel-in-time step is solved by a variant of bound-preserving Newton methods with a nonlinear elimination preconditioner, where the corresponding linear system is solved by the space-time restricted additive Schwarz (stRAS) method. Since the straightforward extension of one-level stRAS to multilevel does not work because of the pollution effects from the temporal direction, we accordingly present a nonstandard V-cycle multilevel stRAS method with a pollution-removing strategy to accelerate the convergence and improve the robustness. Numerical experiments are presented to demonstrate that the aforementioned parallel-in-time algorithm can not only achieve a high degree of parallelism in accurately resolving reservoir transport features in a heterogeneous medium, but also successfully circumvent the convergence and scalability issues associated with the constraint of large time steps and the boundedness requirement of the solution.
Published Version
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