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.
