Construction, Analysis, Implementation and Application of New Time Integrators for Large Scale Complex Systems

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Computer simulations are an essential tool of science and engineering. Relying on computational models to design a car or predict movement of a hurricane is commonplace practice that allows scientists and engineers to significantly reduce time and resources required to accomplish these tasks. The complexity of phenomena we are interested in simulating is constantly increasing. Demand for predictive computer models of complex systems dynamics, such as folding of a protein molecule or movement of a tsunami, is ever-growing and cannot be met with advances in computational hardware alone. Advanced mathematical methods and computational techniques are an integral component of improving fidelity and efficiency of predictive computational models. In particular, numerical methods, which enable simulation of the time evolution of a complex system, are needed to create computer models that can accurately and efficiently predict the behavior of the system over long periods of time. The proposed research focuses on construction of advanced numerical techniques that enable simulating dynamics of complex systems that evolve over a wide spectrum of temporal scales. The project will include both development of theory of such methods as well as their implementation and application to specific problems such as numerical weather prediction and climate modeling. This work will make available software package that can be utilized in a wide variety of scientific and engineering fields ranging from biochemistry and geo-engineering to fluid dynamics and plasma physics. The proposed project will advance the state-of-the-art in time integration from theoretical and practical perspectives. It has become increasingly clear that fast high-order discretization methods to resolve spatial and temporal scales of complex phenomena are essential to ensure that computational models are sufficiently efficient and trustworthy. While extensive research efforts have been dedicated to the development of high-order spatial discretization approaches such as finite-element or spectral methods, research on construction and application of advanced time integrators to large-scale complex problems lags behind. The proposed research will focus on development of new time integration algorithms that significantly improve the efficiency of currently available methods. The project aims to construct, analyze and implement time integration schemes that can simultaneously take advantage of explicit, implicit and exponential integration approaches. Important theoretical questions such as accuracy and convergence of these techniques will be addressed. General integrators as well as classes of methods that can exploit particular problem structure will be derived. In addition, custom-designed time integrators will be created for weather and climate prediction problems to maximize the efficiency of computational models routinely used in these fields. The most efficient time integration schemes developed as a result of this project will be implemented as part of a publicly available software package designed for serial and parallel computational platforms. Outcomes of this research will also be integrated into the core graduate curriculum of the applied mathematics program at UC Merced.