Robust High-Order Methods for Wave Equations in the Time Domain

本研究的目的是解决在发展稳健和有效的计算方法模拟波浪的基本问题。由波传播控制的问题跨越了我们所经历的许多物理现象，并在通信和成像的工程系统以及地球环境的自然发生方面发挥了基本作用。具体的例子包括：与地震和海啸等自然灾害有关的波浪现象；机场附近和城市的声污染等环境刺激物；对国防和民用应用具有重要意义的电磁现象，如雷达成像；以及在医学上的应用，比如高频超声和组织的相互作用。该项目将开发用于模拟波浪的改进工具，并将设计相关的通用开源软件，这些软件在一系列重要应用领域具有重大影响的潜力。随着计算能力的惊人增长，我们期望模拟更困难和更全面的物理现象模型。对于时域中的波传播问题，这意味着在计算域内具有许多波长的问题，涉及与复杂几何特征的相互作用。为了有效地处理这类问题，需要使用高阶离散化方法来最小化分散和耗散的影响。这项工作将集中在进一步开发鲁棒高阶波解算器所需的基本数学问题上。其中包括：1 .结构-非结构混合网格能量稳定高阶/高分辨率离散化方法的开发与分析。具体来说，我们将研究在复杂边界和材料界面附近的非结构化网格上的高阶迎风不连续伽勒金方法与更有效的结构化网格方法的耦合，如基于Hermite-Birkhoff插值的新型谱元方法（也称为射流方案）或由分段多项式或带限插值函数构建的迎风差分方法。一阶和二阶双曲系统都将被考虑。2。这些方法的hp适应性策略的开发、分析和实施。3。与开源辐射边界条件库（预计2014年底发布）耦合，其中包含各种完整辐射边界条件（CRBC）的实现。这些允许先验地确定边界条件参数，以保证远场各向同性均匀模型的任何所需精度。iv.利用crbc对任何弗里德里希系统都是稳定的这一事实，将其适用性扩展到更复杂的物理模型，包括各向异性。

The goal of this research is to address basic issues in the development of robust and efficient computational methods for simulating waves. Problems governed by wave propagation span much of the physical phenomena we experience and play a fundamental role both in engineered systems for communication and imaging as well as naturally-occurring aspects of earth's environment. Specific examples include: wave phenomena involved with natural disasters such as earthquakes and tsunamis; environmental irritants such as acoustic pollution near airports and in cities; electromagnetic phenomena of importance to defense and civilian applications, such as radar imaging; and applications in medicine such as the interaction of high-frequency ultrasound and tissue. This project will develop improved tools for simulating waves and will design associated general-purpose open-source software with the potential for significant impact in a range of important application areas.With the staggering increases in computational power that have been and continue to be achieved, we expect to simulate more difficult and comprehensive models of physical phenomena. For wave propagation problems posed in the time domain, this means problems with many wavelengths within the computational domain involving interactions with complex geometrical features. To treat such problems efficiently requires the use of high-order discretization methods to minimize the effects of dispersion and dissipation. This work will be focused on fundamental mathematical issues required for the further development of robust, high-order wave solvers. These include: i. Development and analysis of energy-stable high-order/high-resolution discretization methods on hybrid structured-unstructured grids. Specifically we will investigate coupling high-order upwind discontinuous Galerkin methods on unstructured grids near complex boundaries and material interfaces with more efficient structured grid methods such as novel spectral element methods based on Hermite-Birkhoff interpolation (also known as jet schemes) or upwind difference methods constructed from piecewise polynomial or band-limited interpolation functions. Both first-order and second-order hyperbolic systems will be considered. ii. Development, analysis, and implementation of hp-adaptive strategies for these methods. iii. Coupling with an open-source radiation boundary condition library (expected release late 2014) containing various implementations of complete radiation boundary conditions (CRBC). These allow a priori determination of the boundary condition parameters to guarantee any desired accuracy for isotropic, homogeneous models in the far field. iv. Leveraging the fact that CRBCs are stable for any Friedrichs system, extend their applicability to more complex physical models including anisotropy.