Asymptotically Improved Solvers for the Helmholtz Equation

亥姆霍兹方程的渐近改进求解器

• 批准号: RGPIN-2021-02613
• 负责人: Bremer, James
• 金额: $ 1.53万
• 依托单位: University of Toronto
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2021
• 资助国家: 加拿大
• 起止时间: 2021-01-01 至 2022-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=741854
• 关键词: Asymptotically; Improved; Solvers; Helmholtz; Equation

The importance of the numerical simulation of physical phenomena by computers cannot be overstated. Such computations have become essential tools both in scientific research and in industrial research and development. This research concerns the numerical simulation of the scattering of waves. Such simulations have applications to sonar and radar, as well as in medical imaging, geophysics, and many other areas. Wave phenomena become more complicated to model as the frequency of the wave increases, and our current ability to accurately model high-frequency waves is quite limited. This project seeks to develop new methods for modeling high-frequency waves efficiently and to high accuracy. The project provides research training opportunities for postdoctoral researchers, graduate students and undergraduates. In many cases of interest, the numerical simulations of the scattering of waves can be performed by solving the variable coefficient Helmholtz equation. The solutions of this equation are oscillatory, and the difficulty of calculating them using conventional approaches grows quickly with the frequency of the oscillations. Recently, the investigator has developed a new class of solvers for the variable coefficient Helmholtz equation that achieve extremely high accuracy and have run times that scale much more slowly with increasing frequency than conventional solvers. They operate by solving the nonlinear Riccati equation that is satisfied by the logarithms of solutions of the Helmholtz equation. Currently, these solvers only apply in special cases. The main thrust of this research program is to extend them to the general case to develop a method for the variable coefficient Helmholtz equation that is significantly faster than current techniques.

计算机对物理现象的数值模拟的重要性不能被夸大。这些计算已成为科学研究和工业研究和发展中的重要工具。这项研究涉及波浪散射的数值模拟。此类模拟对声纳和雷达以及医学成像，地球物理学以及许多其他领域都有应用。随着波浪频率的增加，波浪现象变得更加复杂，并且我们当前准确地对高频波建模的能力非常有限。该项目旨在开发有效地建模高频波和高精度的新方法。该项目为博士后研究人员，研究生和本科生提供了研究培训机会。在许多感兴趣的情况下，可以通过求解可变系数Helmholtz方程来对波的散射进行数值模拟。该方程式的解决方案是振荡性的，并且使用常规方法计算它们的难度随振荡的频率而迅速增长。最近，研究者为可变系数Helmholtz方程开发了一类新的求解器，该方程达到了极高的精度，并且运行时间比传统求解器的频率增加得多。它们通过求解非线性riccati方程来运行，该方程可通过helmholtz方程的溶液对数满足。目前，这些求解器仅适用于特殊情况。该研究计划的主要目的是将其扩展到一般情况下，以开发一种可变系数Helmholtz方程的方法，该方法明显快于当前技术。