Mathematical Sciences: Development and Application of Advanced Numerical Methods to Outstanding Problems in Experimental Shock Waves Geophysics

9316529 Puckett这项研究的中心是设计和开发一种数值方法，用于模拟高压和高温下凝聚态物质中的复杂非线性波相互作用。这种数值方法是基于一系列数值算法，这些算法已成功地用于模拟气体中的冲击波折射。将这些算法扩展到涉及凝聚态的问题，包括修改以适应现实的凝聚相状态方程，包含真实的本构关系来模拟固体中的强度效应，以及开发用于描述真空态的稳健算法。从数值计算中获得的对凝聚态中波动现象的更好的理解也将有助于对波结构的理解，这种结构表征了比控制气体流动的双曲型守恒律组更一般的解，从而有助于对这些方程的数学分析。这项工作包括对数值结果进行广泛的实验验证。该项目致力于开发先进的计算方法，基于最初为解决航空航天和国防问题而开发的方法，并将新方法应用于地球科学中的重要问题。这些新的计算方法被用来设计一种改进的实验技术，用于确定地球物理材料中的声速。准确的声速知识对于正确解释地震数据非常重要。新的计算方法也被用来模拟大型彗星或小行星对地球的影响。这种影响被提出用来解释物种(如恐龙)的大规模灭绝，并可能是地球早期大气周期性丧失的原因。讨论了这种大冲击的具体性质，以及它们可能对大气和地球磁场造成的后果。文中还对计算方法在模拟火山喷发中的应用进行了研究。这项研究开发的计算方法将在其他科学和技术领域有广泛的应用。

9316529 Puckett The central focus of this research is the design and development of a numerical method for modeling complex nonlinear wave interactions in condensed matter at high pressures and temperatures. This numerical method is based on a collection of numerical algorithms that have been used with great success to model shock wave refraction in gases. The extension of these algorithms to problems involving condensed matter involves modifications to accommodate a realistic condensed phase equation of state, the inclusion of realistic constitutive relations for modeling the strength effects in solids, and the development of robust algorithms for describing vacuum states. It is also expected that a better understanding of wave phenomena in condensed matter gained from the numerical computations will contribute to the understanding of the wave structure that characterize solutions of more general systems of hyperbolic conservation laws than those that govern the flow of gases, and thus aid in the mathematical analysis of these equations. This work includes extensive experimental validation of numerical results. This project is concerned with developing advanced computational methods, based on methods originally developed to address aerospace and defense problems, and applying the new methods to important problems in the geosciences. These new computational methods are used to design an improved experimental technique for determining sound speeds in geophysical materials. Accurate knowledge of sound speeds is important for the correct interpretation of seismic data. The new computational methods are also used to model the effect the impact of a large comet or asteroid would have on the Earth. Such impacts have been proposed to explain the mass extinction of species (e.g., dinosaurs) and may have been responsible for the periodic loss of the Earth's early atmosphere. The specific nature of such large shocks, and the consequences they might hav e for the atmosphere and the Earth's magnetic field, are addressed. The use of the computational methods to model explosive volcanic eruptions is also investigated. The computational methods developed in this research will have a wide range of applications in other areas of science and technology.