New Finite Element Techniques for Simulating Flows and Waves

各种自然和技术过程的计算机模拟需要精确和有效的流和波的数值近似。该项目通过开发非常规方法来提高此类模拟的核心数值技术，从而提高了当前的技术水平。除了预期的数学发明外，该项目还计划开发一种实施新技术的公共领域开源软件产品，并使用该产品解决各种应用领域的模拟问题，包括减轻地质灾害和材料科学。通过纳入代表性不足的少数民族，项目活动有助于基金会扩大所有人参与科学的目标。该项目的技术工作分为两条调查线。其中一个导致了捕捉双曲系统的波解的新方法，这些方法有望在新兴的多核架构中脱颖而出。另一项研究利用一种新的数学成分来获得保持粘性流动结构的数值近似。探究的第一行源于这样的观察：当双曲解可以在不同的空间位置通过不同的量在时间上推进时，计算资源的分配得到了优化。当这也可以同时完成时，可以更快地进行模拟。预测的结果是对因果时空帐篷的非结构化时空网格进行快速处理的新方法。该项目的进展有利于各种波传播系统和无粘可压缩流的模拟。粘性不可压缩流动是第二项研究的目标。利用矩阵值函数的Sobolev空间，构建了新的公式，产生最佳的流体应力近似，精确的质量守恒和压力稳健性。这两种调查方式的工作都涉及回答几个技术问题，这些问题被证明是基本的，并且可能超出本项目范围的学科影响。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Computer simulations of various natural and technological processes require accurate and efficient numerical approximation of flows and waves. This project advances the state of the art by developing unconventional approaches that improve numerical techniques at the heart of such simulations. In addition to the anticipated mathematical inventions, the project also plans to develop a public-domain open-source software product implementing the new techniques and use the product to solve simulation problems in varied application domains including geological hazard mitigation and material science. Through inclusion of under-represented minorities, the project activities contribute to the foundation's goals to widen participation of all in science.The technical work on the project is divided into two lines of inquiry. One leads to new methods for capturing wave solutions of hyperbolic systems, methods that are expected to excel on the emerging many-core architectures. Another line of inquiry leverages a new mathematical ingredient to obtain structure-preserving numerical approximations of viscous flows. The first line of inquiry is motivated by the observation that when hyperbolic solutions can be advanced in time by varying amounts at varying spatial locations, the allocation of computational resources is optimized. When this can also be done concurrently, much faster simulations are possible. New fast methods on unstructured spacetime meshes of causal spacetime tents are the projected outcome. The advances from this project benefit simulation of various wave propagation systems as well as inviscid compressible flows. Viscous incompressible flows are targeted by the second line of inquiry. Using a Sobolev space of matrix-valued functions, novel formulations are constructed that yield optimal fluid stress approximations, exact mass conservation, and pressure robustness. Work in both lines of inquiry involve answering several technical questions, shown to be fundamental, and of potential disciplinary impact beyond the confines of this project.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.