Construction of New Parallel Time Integrators

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For many problems in science and engineering experimental data and observations do not provide a complete picture of a system's behavior and computer simulations have become an indispensable tool for studying its evolution. However, the complexity and scale of many application problems make even computer modeling a challenging task. This is particularly true when a system undergoing complex dynamics is modeled using time dependent partial differential equations. In fact, special mathematical techniques known as time integrators must be developed to ensure that the evolution of the system is reproduced on a computer with a reasonable fidelity and in a reasonable time. It is also important that such time integration algorithms have the ability to take advantage of the parallel computational capabilities offered by both supercomputers and workstations. This project will develop new time integration methods that exploit parallelism and novel mathematical ideas in order to achieve higher accuracy and better efficiency when simulating complex systems that evolve over a wide range of temporal and spatial scales. General methods applicable to problems across science and engineering will be developed as well as special schemes particularly optimized for certain applications. The project will explore the advantages of the new numerical approaches in the context of real-world problems such as computer design of efficient engines and weather prediction. This project will also contribute to educating and mentoring undergraduate and graduate students.Employing parallelism effectively in the spatial discretization of partial differential equations has proven to be revolutionary in computational science. Recent developments in time integration provide an opportunity to extend these gains to the time domain. This project will combine ideas from exponential and polynomial interpolation-based temporal integration to construct new parallel time integrators. Both parallelization across the method (i.e. over each time iteration) and parallelization across time steps (i.e. simultaneous computation of the solution over multiple time intervals) will be explored. New approaches for fine-tuning the parameters to optimize performance of the new methods will also be investigated. The behavior of the new methods will be studied in context of real application problems such as modeling reactive flows involved in combustion. An additional objective of this work is to provide a guide for determining which of the new techniques are best suited for problems with a given structure.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.