Numerical Methods for Conservation Laws

双曲守恒律系统出现在许多问题中，这些问题中的物理量，如质量，动量和能量的一致性被建模。这些方程通常必须用近似数值方法求解。 数据和/或解决方案的不连续性（例如冲击波）导致难以开发准确和稳定的方法。该建议涉及高分辨率方法的发展及其作为软件的实现，可广泛用于解决许多不同领域中出现的问题。 私家侦探一直积极参与开发三维双曲方程组的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各种现象，使用高分辨率方法和自适应网格细化技术。 该软件将进一步开发，并用于探索科学和工程中的一些特殊应用。 一个应用是计算非均匀介质中的声波或弹性波。 例如，这样的问题经常出现在生物学和材料科学中。 另一个应用是数值相对论，人们对模拟可能很快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线公式来实现。

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.