U.S.-Czech Mathematics Research on Symbolic Derivation, Analysis, and Programming of Finite Difference Schemes

这个关于有限差分格式的符号推导、分析和规划的美国-捷克数学研究项目是由新墨西哥大学的stanley Steinberg博士和布拉格捷克技术大学的Richard Liska博士共同完成的。他们的研究成果将扩展目前用于研究有限差分近似的符号技术。研究人员希望改进现有的分析截断误差和稳定性的技术，以及生成代码的算法。圆柱分解算法将用于研究由冯·诺依曼稳定性条件得到的劳斯-赫尔维茨问题中出现的不等式系统。将发展在非正交网格上比较有限差分算法的技术。研究人员将继续研究数值扩散和色散、群速度、边界条件的稳定性、守恒定律以及通过能量方法进行稳定性分析。由于他们的合作，研究人员希望将fortran代码生成算法扩展到高性能计算环境中，用于研究五维Fokker-Planck方程和三维Navier Stokes方程。​该数学项目通过使美国和东欧的顶尖专家在共同感兴趣和能力强的领域结合互补人才和集中研究资源，实现了推进科学知识的计划目标。

This US-Czech mathematics research project on Symbolic Derivation, Analysis, and Programming of Finite Difference Schemes is between Dr. Stanly Steinberg of the University of New Mexico and Dr. Richard Liska of the Czech Technical University, Prague. Their research effort will extend the current symbolic techniques used to study finite-difference approximations. The researchers hope to improve the existing technology for analyzing truncation errors and stability, and the algorithms for generating code. Cylindrical decomposition algorithms will be used to study a system of inequalities that arise in a Routh- Hurwitz problem obtained from the Von Neumann stability condition. Techniques for comparing finite-diffence algorithms on non-orthogonal grids will be developed. The researchers will pursue work with numerical diffusion and dispersion, group velocities, stability of boundary conditions, conservation laws, and stability analysis via the energy method. As a result of their collaboration, the researchers expect to extend the FORTRAN-code generation algorithms to a high-performance computing environment for use in study of a five- dimensional Fokker-Planck equation and the three-dimensional Navier Stokes equation. This project in mathematics fulfills the program objective of advancing scientific knowledge by enabling leading experts in the United States and Eastern Europe to combine complimentary talents and pool research resources in areas of strong mutual interest and competence.