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.  

　　Publication:  

　　Journal of Computational Physics, Volume 494, 1 December 2023, 112515  

　　http://dx.doi.org/10.1016/j.jcp.2023.112515  

　　Author:  

　　Tianpei Cheng  

　　School of Mathematics, Hunan University, Changsha, Hunan 410082, PR China  

　　Haijian Yang  

　　School of Mathematics, Hunan University, Changsha, Hunan 410082, PR China  

　　Jizu Huang  

　　Institute of Computational Mathematics and Scientific/Engineering Computing, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, PR China  

　　Email: huangjz@lsec.cc.ac.cn