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

时域波动方程的鲁棒高效数值方法：非线性和多尺度问题

• 批准号: 2309687
• 负责人: Thomas Hagstrom
• 金额: $ 39万
• 依托单位: Southern Methodist University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-07-01 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2309687&HistoricalAwards=false
• 关键词: Robust; Efficient; Numerical; Methods; Wave

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.

该项目旨在开发快速可靠的算法来模拟波，能够利用当前和未来的计算技术以及处理广泛的物理系统数学模型。波浪现象在物理科学和工程的几乎所有领域都很重要。它们对于传感和成像应用，对潜在的自然灾害（例如地震）的建模以及人类对了解宇宙的追求的建模至关重要。从长远来看，开发的算法将被纳入软件以促进其广泛使用。除了对解决波浪物理学的全面模型的基本工具的贡献外，该项目还将培训三名计算科学领域的研究生，部分作为资助的研究助理，并进一步发展研究和教育中跨学科计划，涉及计算作为重要组成部分。波浪理论中问题提出的计算技术的主要挑战植根于通常发生的多个时间和空间尺度。波波在没有明显衰减的许多波长上传播的基本事实导致数值方法必须具有最小的分散和耗散误差，并且必须使用准确的近似辐射边界条件限制计算域。另外，波浪可能与复杂的几何对象相互作用，并具有自己的固有长度尺度。该项目将解决所有这些问题。基于先前成功开发统一培养基中线性双曲线问题的最佳合理近似值，将使用降低的订单建模算法来构建有效的方法，以解决库仑电位的问题，在定期媒体，多尺度介质中以及Schrodinger-type方程。研究人员将提高其离散化方法的效率，由于能量稳定性，它们具有鲁棒性，具有用于平滑溶液的任意收敛顺序，并且能够处理由动作原理引起的一般二阶双曲系统。具体来说，这项研究将开发高阶本地时间步变的计划，以更有效地处理本地精致和混合网格，并将在包括引力和拓扑浪潮在内的有趣的物理问题上行使他们的方法。该奖项反映了NSF的法定任务，并通过使用该基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的支持。