An Adaptive Newton–ASPEN Solver for Complex Reservoir Models

Standard Newton methods that are used to advance fully implicit or adaptive implicit schemes in time often suffer from slow or stagnant convergence when natural initial guesses are too far from the solution or the discrete flow equations contain nonlinearities that are unbalanced in time and space. Nonlinear solvers based on local-global, domain-decomposition strategies have proved to be significantly more robust than regular Newton but come at a higher computational cost per iteration. The chief example of one such strategy is additive Schwarz preconditioned exact Newton (ASPEN) that rigorously couples local solves, which in sum have little cost compared with a Newton update, with a global update that has a cost comparable to a regular Newton solve. We present strategies for combining Newton and ASPEN to accelerate the nonlinear solution process. The main feature is a set of novel monitoring strategies and systematic switching criteria that prevent oversolving and enable us to optimize the choice of solution strategy. At the start of each nonlinear iteration, convergence monitors are computed and can be used to choose the type of nonlinear iteration to perform as well as methods, tolerances, and other parameters used for the optional local domain solves. The convergence monitors and switching criteria are inexpensive to compute. We observe the advantages and disadvantages of local-global domain decomposition for practical models of interest for oil recovery and CO2 storage and demonstrate how the computational runtime can be (significantly) reduced by adaptively switching to regular Newton's method when nonlinearities are balanced throughout the physical domain and the local solves provide little benefit relative to their computational cost.