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）应用数学组的Rolf Jeltsch教授在数学研究方面的合作。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找到，如有限元法、谱法和有限差分法。在这个建议中，强调有限差分方法。计算流体力学、地球物理和空气动力学等许多领域都需要双曲型初边值问题的现实解。一个完整的几何描述的黎曼曲面的顺序星将提供洞察的稳定性和精度的数值算法来解决这些问题。两位研究者在双曲型方程的数值解方面都很活跃；雷诺特博士的工作主要集中在波动方程上，而耶尔施博士的工作主要集中在平流方程上。对这些数值算法的深入理解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.