Robust and Efficient Numerical Methods for Wave Equations in the Time Domain: Nonlinear and Multiscale Problems

该项目旨在开发快速可靠的海浪模拟算法，能够利用当前和未来的计算技术，并处理各种物理系统的数学模型。波动现象在几乎所有的物理科学和工程领域都很重要。它们是传感和成像应用的核心，对地震等潜在自然灾害的建模，以及对人类了解宇宙的探索。从长远来看，开发的算法将被整合到软件中，以促进其广泛应用。除了对解决波动物理综合模型的基本工具作出贡献外，该项目还将培训三名计算科学研究生，部分作为资助的研究助理，并进一步发展将计算作为重要组成部分的跨学科研究和教育计划。波动理论中的问题给计算技术带来的主要挑战源于通常发生的多个时间和空间尺度。一个波在许多波长上传播而没有显著衰减的基本事实导致要求数值方法必须具有最小的色散和耗散误差，并且必须使用精确的近似辐射边界条件来限制计算域。此外，波可以与具有其固有长度尺度的复杂几何对象相互作用。该项目将解决所有这些问题。在前人研究均匀介质中线性双曲问题辐射边界条件最优有理逼近的基础上，降阶建模算法将被用于构造求解库仑势问题、周期介质问题、多尺度介质问题和薛定谔方程的有效方法。研究人员将提高他们的离散化方法的效率，这些方法由于能量的稳定性而具有健壮性，具有光滑解的任意收敛阶，并且能够处理由作用原理产生的一般二阶双曲组。具体地说，这项研究将开发更高阶的局部时间步长方案，以更有效地处理局部细化和混合网格，并将在有趣的物理问题上应用它们的方法，包括引力波和拓扑波。这一奖项反映了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。

The project aims to develop fast and reliable algorithms for simulating waves, capable of exploiting current and future computing technology and of treating a wide range of mathematical models of physical systems. Wave phenomena are important in almost all areas of the physical sciences and engineering. They are central to sensing and imaging applications, to the modeling of potential natural disasters such as earthquakes, and to humanity's quest to understand the universe. In the longer term, the algorithms developed will be incorporated into software to promote their broad use. Besides contributions to basic tools for solving comprehensive models of wave physics, the project will result in the training of three graduate students in computational science, supported in part as funded research assistants, and in the further development of transdisciplinary programs in research and education which involve computation as an important component. The main challenges to computational technique posed by problems in wave theory are rooted in the multiple temporal and spatial scales which typically occur. The fundamental fact that a wave propagates over many wavelengths without significant attenuation leads to the requirement that numerical methods must have minimal dispersion and dissipation errors and that the computational domain must be limited using accurate approximate radiation boundary conditions. In addition, the waves may interact with complex geometrical objects with their own inherent length scales. The project will address all of these issues. Building on previous success in developing optimal rational approximations to radiation boundary conditions for linear hyperbolic problems in uniform media, reduced order modeling algorithms will be used to construct effective methods for problems with Coulomb potentials, in periodic media, in multiscale media, and for Schrodinger-type equations. The investigator will improve the efficiency of their discretization methods, which are robust, due to energy stability, have arbitrary convergence orders for smooth solutions, and are capable of treating general second-order hyperbolic systems arising from action principles. Specifically, the research will develop higher order local time-stepping schemes to more efficiently treat locally refined and hybrid meshes and will exercise their methods on interesting physical problems, including gravitational and topological waves.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.