Hybrid Hermite-Discontinous Galerkin Methods with Applications to Elastic and Electromagnetic Waves

首席研究员建议开展一项跨学科的综合研究计划，将一类新的数值方法的开发、分析和优化与它们在地震学和电磁学问题中的应用结合起来。新方法将任意阶Hermite近似与任意阶间断Galerkin方法相结合。这两种方法的结合将产生一类新的混合方法，能够处理复杂的几何图形，并通过大时间步长和高分辨率获得前所未有的计算效率。这些方法具有很大的计算与通信比，非常适合在现有和新兴的超级计算机系统上实现，从而能够解决复杂的、多尺度的演化系统。提出的间断Galerkin方法和Hermite方法的统一分析将需要开发新的工具和理论，并将导致分析混合方法的新的理论框架。该提案将考虑弹性力学和电磁学控制方程的一阶和二阶公式的方法。这项研究将在技术和科学以及对下一代计算科学家的培训方面产生更广泛的影响。正如最近在日本发生的事件所表明的那样，地震是全世界的一个社会问题。为了更好地减轻地震灾害，需要有效的预防和预测。通过准确的计算预测对地震危险性进行仔细的评估可以产生适当的建筑规范。在大洛杉矶或旧金山湾等人口稠密地区发生大地震时，这可能对人类生活和社会福利产生巨大影响。拟议活动的更广泛影响还包括教育。该项目将涉及研究生，他们将获得最先进的计算科学方面的经验。这项研究将在新墨西哥大学进行，这是一家服务于拉美裔美国人的机构，也为大量美国原住民提供服务，允许积极招收和教育来自代表性不足群体的学生。

The Principal Investigator proposes to carry out an interdisciplinary comprehensive research program combining the development, analysis and optimization of a new class of numerical methods, with their application to problems in seismology and electromagnetics. The novel methods hybridizes arbitrary-order Hermite approximations with arbitrary-order discontinuous Galerkin methods. The combination of these two methods will result in a new class of hybrid methods able to handle complex geometries and with unprecedented computational efficiency through large time steps and high-resolution. The methods have very large computation to communication ratio and are well suited for implementation on current and emerging supercomputer systems, enabling the solution of complex, multiple-scale evolutionary systems. The proposed unified analysis of discontinuous Galerkin methods and Hermite methods will require new tools and theories to be developed and will lead to a new theoretical framework for the analysis of hybrid methods. The proposal will consider methods for both first and second order formulations of the governing equations of elasticity and electromagnetics. The research will have broader impacts in technology and science, as well as in the training of the next generation of computational scientists. As recent events in Japan have shown, earthquakes are a societal problem throughout the world. To better mitigate seismic hazard, effective prevention and prediction is needed. Careful assessment of seismic hazards through accurate computational predictions can lead to appropriate building codes. This can be of enormous impact for human life and societal welfare in the case of a large seismic event in a densely populated area as the greater Los Angeles or the San Francisco bay. The broader impacts of the proposed activities also include education. The project will involve graduate students who will gain experience in state-of-the-art computational science. The research will be performed at the University of New Mexico, a Hispanic serving institution that also serves a large body of native Americans, allowing active recruitment and education of students from underrepresented groups.