U.S.-Switzerland Cooperative Research on Order Stars, Riemann Surfaces, and Implicit Solutions of Hyperbolic Problems (Applied Mathematics)

该奖项支持亚利桑那州立大学的Rosemary Renaut教授与苏黎世瑞士联邦研究所(ETH)应用数学小组的Rolf Jeltsch教授在数学研究方面进行合作。他们正在研究求解双曲型微分方程的某些数值算法的稳定性。通常，这些微分方程的数值解可以通过各种方法来找到，如有限元、谱方法和有限差分方法。在该方案中，重点介绍了有限差分方法。双曲型初边值问题在计算流体力学、地球物理和空气动力学等许多领域都需要现实的解。黎曼曲面阶星的完整几何描述将提供对解决这些问题的数值算法的稳定性和精度的洞察。这两位研究人员都活跃在双曲型方程的数值解上；雷诺博士的工作主要集中在波动方程上，而耶尔奇博士则致力于平流方程。更深入地理解这些数值算法将使许多其他问题受益。

This award supports Professor Rosemary Renaut of Arizona State University to collaborate in mathematics research with Professor Rolf Jeltsch of the Applied Mathematics Group of the Swiss Federal Institute (ETH), Zurich. They are studying the stability of certain numerical algorithms for solving hyperbolic differential equations. Typically, numerical solutions to these differential equations can be found by a variety of methods such as finite elements, spectral methods and finite difference methods. In this proposal finite difference methods are emphasized. Realistic solutions of hyperbolic initial boundary value problems are required in many areas including computational fluid mechanics, geophysics and aerodynamics. A complete geometric description of the order star of Riemann surfaces will provide insight into the stability and accuracy of numerical algorithms to solve these problems. Both researchers have been active in the numerical solution of hyperbolic equations; Dr. Renaut's work has been mainly concentrated on the wave equation, while Dr. Jeltsch has worked on the advection equation. Many other problems will benefit from a deeper understanding of these numerical algorithms.