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

在这个项目中，PI开发了算法和软件，以方便通过高阶近似方法（如伪谱和径向基函数方法）从不连续数据的近似中去除吉布斯振荡。目前的后处理技术在一维上是比较成功的。然而，最强大的方法需要精确定位不连续点（边缘）的位置，并在每个分段光滑区域中指定函数相关参数。工作的一个重点是开发无边缘检测（EDF）后处理方法。EDF方法很容易扩展到多个维度，在某些情况下可以扩展到不规则形状的域。最终目标是一个“黑盒”后处理算法。第二个工作重点是软件开发。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，为专家和非专业人员提供方法的说明，并刺激进一步的研究。该软件旨在使学生相对容易地学习这些方法，并使研究人员能够应用这些方法。要想用最精确的方法来数值逼近支配自然现象的方程，后处理方法至关重要。这是因为自然界中的许多现象本质上是不连续的，当控制方程的解是不连续的时候，数值方法就不能很好地发挥作用。这种不连续性导致产生非物理振荡的解的方法不能真正代表被建模的现象。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、天体物理建模、多相流体流动、冰川流动、波在弹性固体中的传播、化学物质的分离、空气动力学、爆震波、声学等。

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.