Computational scientists are grappling with increasingly complex and coupled physics in applications such as fluid dynamics, fusion, electronics, groundwater flow, astrophysics, and combustion, where individual effects were previously studied. Moreover, parallel computers with large storage capacities have paved the way for high-resolution simulations of large-scale problems. The growth in complexity and size of models has led to an increased demand for solution methods for large-scale, time-dependent, nonlinear problems. To accurately capture nonlinear coupling between dynamically relevant phenomena while stepping over fast waves or rapid adjustments to quasi-equilibria, simulation scientists are increasingly turning to implicit formulations which often result in a coupled, nonlinear system to be solved at each time step. Recent advances in iterative methods have made fully implicit formulations a viable option for solution of these problems. In this talk, we will overview the most effective iterative methods and algorithmic choices for nonlinear systems and point to software with their implementation. We will then give examples of these methods within scientific applications, including fusion, astrophysics, and groundwater flow, and discuss current research areas in extending their use. Lastly, we will overview algorithm extensions which can result in further scientific insight.
This work was performed under the auspices of the U.S. Department of Energy by University of California Lawrence Livermore National Laboratory under contract No. W-7405-Eng-48.