Numerical simulation of water resources problems: Models, methods, and trends

► Evolving water resource model formulations are reviewed. ► Trends in solution algorithms are considered. ► Recent advances and challenges in discretization methods and solvers are presented. ► Computational tools, trends, and implications are discussed. Mechanistic modeling of water resources systems is a broad field with abundant challenges. We consider classes of model formulations that are considered routine, the focus of current work, and the foundation of foreseeable work over the coming decade. These model formulations are used to assess the current and evolving state of solution algorithms, discretization methods, nonlinear and linear algebraic solution methods, computational environments, and hardware trends and implications. The goal of this work is to provide guidance to enable modelers of water resources systems to make sensible choices when developing solution methods based upon the current state of knowledge and to focus future collaborative work among water resources scientists, applied mathematicians, and computational scientists on productive areas.