RUI: Postprocessing High-Order Approximations of Discontinuous Functions - Algorithms and Software

RUI：不连续函数的后处理高阶近似 - 算法和软件

• 批准号: 0609747
• 负责人: Scott Sarra
• 金额: $ 7.32万
• 依托单位: Marshall University Research Corporation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-15 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0609747&HistoricalAwards=false
• 关键词: RUI; Postprocessing; Order; Approximations; Discontinuous

In this project the PI develops algorithms and software tofacilitate the removal of Gibbs oscillations from the approximationof discontinuous data by high order approximation methods such aspseudospectral and radial basis function methods. Currentpostprocessing techniques are somewhat successful in one dimension.However, the most powerful methods require that the exact locationof the discontinuities (edges) be pinpointed and that functiondependent parameters be specified in each piecewise smooth region.One focus of the work is in developing Edge Detection free (EDF)postprocessing methods. The EDF methods are easily extended tomultiple dimensions and in some case to irregularly shaped domains.The ultimate goal is a "black box" postprocessing algorithm. Asecond focus of the work is in software development. A Matlabtoolbox which implements both classical and modern postprocessingmethods will be developed. The software will include a user friendlygraphical user interface facilitating use by the wider community.The software will be used on benchmark problems to provideillustration of the methods for both the specialist andnonspecialist and to stimulate further research. The software isdesigned to make it relatively easy for students to learn about themethods and for researchers to apply them.For the most accurate methods available to numerically approximatethe equations that govern natural phenomena, postprocessing methodsare vital. This is because many phenomena in nature are inherentlydiscontinuous, and numerical methods do not perform well when thesolution to the governing equations are discontinuous. Thediscontinuities result in the methods producing solutions withnonphysical oscillations that do not truly represent the phenomenabeing modeled. Areas for which postprocessing methods are importantinclude: gas dynamics, weather forecasting, earthquake and tsunamiprediction, astrophysical modeling, multiphase fluid flow, the flowof glaciers, the propagation of waves in elastic solids, theseparation of chemical species, aerodynamics, detonation waves,acoustics, etc.

在这个项目中，PI开发了算法和软件，以便于去除吉布斯振荡的近似不连续数据的高阶近似方法，如asprodospectral和径向基函数方法.目前的后处理技术在一维图像处理中取得了一定的成功，但大多数后处理方法都要求精确定位不连续点（边缘），并在每个分段光滑区域中指定函数相关的参数，因此，发展无边缘检测（EDF）的后处理方法是研究的重点之一。EDF方法很容易扩展到多维，在某些情况下，不规则形状的域。最终的目标是一个“黑箱”后处理算法。 工作的第二个重点是软件开发。将开发一个实现经典和现代后处理方法的Matlab工具箱。该软件将包括一个用户友好的图形用户界面，方便更广泛的社区使用。该软件将用于基准问题，为专家和非专家提供方法说明，并刺激进一步的研究。 该软件的设计目的是使学生学习这些方法和研究人员应用这些方法相对容易。为了获得最精确的方法来数值近似控制自然现象的方程，后处理方法是至关重要的。这是因为自然界中的许多现象本质上是不连续的，当控制方程的解不连续时，数值方法不能很好地执行。不连续性导致方法产生非物理振荡的解决方案，不能真正代表建模的现象。后处理方法很重要的领域包括：气体动力学、天气预报、地震和海啸预测、天体物理模拟、多相流体流动、冰川流动、弹性固体中波的传播、化学物质的分离、空气动力学、爆炸波、声学等。