How Fast Is Your Newton-Like Nonlinear Solver?

This work answers the question for any Newton-like solver that is applied to nonlinear residual systems arising during the course of implicit Reservoir Simulations. We start by developing a mathematical foundation that characterizes the asymptotic convergence rate of infinite dimensional Newton methods applied to continuous form reservoir simulation problems. Using the fact that finite dimensional (discretized) methods are related to their infinite dimensional counterparts through the approximation accuracy of the underlying numerical discretization scheme, we translate the infinite dimensional characterizations to the finite dimensional world. The analysis reveals the asymptotic scaling relations between nonlinear convergence rate and time-step and mesh size. In particular, we show a constant scaling relation for elliptic problems, a set of super-linear relations for hyperbolic situations, and for mixed parabolic problems. Numerical examples are used to illustrate the theoretical results, and we compare the direct convergence results from this work to those obtained using existing convergence monitoring methods. This work should be of interest to any simulation practitioner or developer who previously relied on text-book quadratic local convergence rate characterizations that did not hold in simulation practice and that perhaps are never even observed. The practical applications of this work are in time-step selection for convergence, generalizing single cell safeguarding tactics, and building insight into asymptotic acceleration methods.