Numerical Methods for Conservation Laws

在对质量、动量和能量等物理量的守恒进行模型化的许多问题中，双曲型守恒定律都会出现。这些方程通常必须用近似数值方法来求解。数据和/或解决方案(例如冲击波)中的不连续导致难以开发准确和稳定的方法。这项建议涉及高分辨率方法的发展及其作为软件的实现，该软件可广泛用于解决许多不同领域中出现的问题。P.I.积极参与开发用于多维双曲型系统的CLAWPACK和AMRCLAW软件。该软件是免费提供的，允许学生和研究人员研究广泛的现象，使用高分辨率方法和自适应网格细化技术。该软件将被进一步开发和使用，以探索科学和工程中的一些特殊应用。其中一个应用涉及计算非均匀介质中的声波或弹性波。例如，这类问题经常出现在地球物理和材料科学中。另一个应用是音量相对论，人们对模拟可能很快就能被天文学家探测到的引力波很感兴趣。这可以通过使用爱因斯坦方程的双曲型公式来实现。

Hyperbolic systems of conservation laws arise in many problems where theconservation of physical quantities such a mass, momentum, and energy aremodeled. These equations must typically be solved by approximate numericalmethods. Discontinuities in the data and/or solution (e.g. shock waves)cause difficulties in developing accurate and stable methods. This proposal concerns the development of high-resolution methods and theirimplementation as software that can be widely used in solving problemsthat arise in many different fields. The P.I. has been activelyinvolved in developing the CLAWPACK and AMRCLAW software formulti-dimensional hyperbolic systems. This software is freelyavailable and allows students and researchers studying a wide range ofphenomena to use the technology of high-resolution methods and adaptivemesh refinement. This software will be further developed and used toexplore a number of particular applications in science andengineering. One application concerns computing acoustic or elasticwaves in heterogeneous media. Such problems often arise in geophysicsand materials science, for example. Another application is tonumerical relativity, where there is interest in simulatinggravitational waves that may soon be detectable by astronomers.This can be accomplished by using a hyperbolic formulation of the Einsteinequations.