Construction of New Parallel Time Integrators

对于科学和工程中的许多问题，实验数据和观测并不能提供系统行为的完整图景，而计算机模拟已经成为研究系统演化的不可或缺的工具。然而，许多应用问题的复杂性和规模使得甚至计算机建模都是一项具有挑战性的任务。当使用依赖于时间的偏微分方程式对经历复杂动态的系统建模时，这一点尤其正确。事实上，必须开发被称为时间积分器的特殊数学技术，以确保在计算机上以合理的保真度和合理的时间再现系统的演变。同样重要的是，这样的时间积分算法具有利用超级计算机和工作站提供的并行计算能力的能力。该项目将开发新的时间积分方法，利用并行性和新颖的数学思想，以便在模拟在广泛的时间和空间尺度上演变的复杂系统时实现更高的精度和更高的效率。将开发适用于跨科学和工程问题的一般方法，以及特别针对某些应用进行优化的特殊方案。该项目将在真实世界问题的背景下探索新的数值方法的优势，例如高效发动机的计算机设计和天气预报。该项目还将有助于教育和指导本科生和研究生。在偏微分方程的空间离散中有效地利用并行性已被证明是计算科学中的革命性。时间积分的最新发展为将这些成果扩展到时间域提供了机会。该项目将结合基于指数和多项式插值的时间积分的思想来构建新的并行时间积分器。将探索跨方法的并行化(即，在每个时间迭代上)和跨时间步长的并行化(即，在多个时间间隔上同时计算解)。还将研究微调参数以优化新方法性能的新方法。新方法的行为将在实际应用问题的背景下进行研究，例如对燃烧中涉及的反应流进行建模。这项工作的另一个目标是为确定哪些新技术最适合于给定结构的问题提供指导。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.