Rami Younis

Adaptively Localized Continuation-Newton Method--Nonlinear Solvers That Converge All the Time

Summary Growing interest in understanding, predicting, and controlling advanced oil-recovery methods emphasizes the importance of numerical methods that exploit the nature of the underlying physics. The fully implicit method offers unconditional stability of the discrete approximations. This stability comes at the expense of transferring the inherent physical stiffness onto the coupled nonlinear residual equations that are solved at each timestep. Current reservoir simulators apply safeguarded variants of Newton’s method that can neither guarantee convergence nor provide estimates of the relation between convergence rate and timestep size. In practice, timestep chops become necessary and are guided heuristically. With growing complexity, such as in thermally reactive compositional flows, convergence difficulties can lead to substantial losses in computational effort and prohibitively small timesteps. We establish an alternative class of nonlinear iteration that converges and associates a timestep to each iteration. Moreover, the linear solution process within each iteration is performed locally. By casting the nonlinear residual equations for a given timestep as an initial-value problem, we formulate a continuation-based solution process that associates a timestep size with each iteration. Subsequently, no iterations are wasted and a solution is always attainable. Moreover, we show that the rate of progression is as rapid as that for a convergent standard Newton method. Moreover, by exploiting the local nature of nonlinear wave propagation typical to multiphase-flow problems, we establish a linear solution process that performs computation only where necessary. That is, given a linear convergence tolerance, we identify a minimal subset of solution components that will change by more than the specified tolerance. Using this a priori criterion, each linear step solves a reduced system of equations. Several challenging examples are presented, and the results demonstrate the robustness and computational efficiency of the proposed method.

Lazy K-Way Linear Combination Kernels for Efficient Runtime Sparse Jacobian Matrix Evaluations in C++

The most notoriously expensive component to develop, extend, and maintain within implicit PDAE-based predictive simulation software is the Jacobian evaluation component. While the Jacobian is invariably sparse, its structure and dimensionality are functions of the point of evaluation. The application of Automatic Differentiation to develop these tools is highly desirable. The challenge presented is in providing implementations that treat dynamic sparsity efficiently without requiring the developer to have any a priori knowledge of sparsity structure. Under the context of dynamic sparse Operator Overloading implementations, we develop a direct sparse lazy evaluation approach. In this approach, an efficient runtime variant of the classic Expression Templates technique is proposed to support sparsity. The second aspect is the development of two alternate multi-way Sparse Vector Linear Combination kernels that yield efficient runtime sparsity detection and evaluation.

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.