A numerical study of the additive Schwarz preconditioned exact Newton method (ASPEN) as a nonlinear preconditioner for immiscible and compositional porous media flow

Abstract

Domain decomposition methods are widely used as preconditioners for Krylov subspace linear solvers. In the simulation of porous media flow there has recently been a growing interest in nonlinear preconditioning methods for Newton’s method. In this work, we perform a numerical study of a spatial additive Schwarz preconditioned exact Newton (ASPEN) method as a nonlinear preconditioner for Newton’s method applied to both fully implicit or sequential implicit schemes for simulating immiscible and compositional multiphase flow. We first review the ASPEN method and discuss how the resulting linearized global equations can be recast so that one can use standard preconditioners developed for the underlying model equations. We observe that the local fully implicit or sequential implicit updates efficiently handle the local nonlinearities, whereas long-range interactions are resolved by the global ASPEN update. The combination of the two updates leads to a very competitive algorithm. We illustrate the behavior of the algorithm for conceptual one and two-dimensional cases, as well as realistic three dimensional models. A complexity analysis demonstrates that Newton’s method with a fully implicit scheme preconditioned by ASPEN is a very robust and scalable alternative to the well-established Newton’s method for fully implicit schemes.

Automatic Time Step Control to Resolve Hydromechanically Driven Fault Reactivation, Spontaneous Nucleation, and Seismic Arrest

“…For instance, during dynamic rupture, the fluid state fields exhibit changes that are localized to the vicinity of traveling mechanical waves. This behavior can be exploited to reduce the computational cost; for example, by using adaptive solution methods (e.g., Klemetsdal, Moncorgé, et al., 2021; Sheth & Younis, 2017; Sheth et al., 2020), mesh refinement methods (e.g., Klemetsdal, Møyner, et al., 2021), and sequential coupling between elastodynamic and poroelastic models.…”

Comparison of nonlinear field-split preconditioners for two-phase flow in heterogeneous porous media

“…The multiplicative Schwarz method has been used to split boundary value problems (BVP) into subproblems solver on smaller physical domains [29][30][31][32][33]. It has also been used to split a coupled BVP into subproblems based on the physics [34][35][36], each subproblem being solved on the full domain to update one of the fields (here, pressure and saturation).…”

“…Two distinct directions have been explored. A class of nonlinear preconditioners leverages domain decomposition methods [29][30][31][32][33] to precondition the nonlinear system in a computationally inexpensive way and speed up convergence. In this work, we focus on nonlinear preconditioners obtained by splitting the system by physical field [34][35][36].…”

On Global and Monotone Convergence of the Preconditioned Newton’s Method for Some Mildly Nonlinear Systems

“…Nonlinear preconditioning appears to be particularly efficient in the application to models of subsurface flow and reactive transport [9], [12], where the failures and lack of robustness of nonlinear solvers is one the major factors limiting the reliability of the simulation codes. In those applications, the major benefit seems to result in the form of an extended convergence region, which, in case of time-dependent problems, allows for larger time steps [12].…”