Further Study of Hierarchical Reconstruction Algorithms

用于求解守恒律的高分辨率捕获格式，例如ENO格式，在几个网格单元内涂抹不连续，并在解是光滑的情况下获得高精度。Cockburn和Shu等人的系列作品。关于不连续Galerkin(DG)方法和局部DG，DG家族引入了许多新技术，使其能够求解包括守恒律在内的更广泛的一类方程。然而，DG的限制技术还不是很成熟，被认为是科学计算中的主要开放问题之一。这位研究人员和他的同事们建议进一步研究一种新的限制技术--分层重建(HR)。这是一个通用的重建过程，用作限制器，以消除在存在冲击时的虚假振荡。HR算法受Biswas，Devine和Flaherty(1994)的矩限制器的启发，在多层重建过程的每个阶段只涉及一个肌肉、一个二阶ENO或其他分段线性重建，而不需要进行特征分解。因此，对于任意网格，该算法结构紧凑，易于实现。它不截断多项式的高次项，而是实际使用所有次数项的信息。证明了HR不降低多项式的逼近阶。此外，HR也可以用于有限体积格式和中心格式，而不需要特征分解，这导致了一种新的有限体积方法。这位研究人员和他的同事们还提出了多相流体模拟中速度平流的局部恒速误差补偿和校正方法(BFECC)、移动网格和网格间内插的BFECC方法的研究。本项目致力于研究和开发利用计算机来模拟某些自然现象的新方法，如气流通过机翼、冲击波在体内的传播、烟雾等。计算机模拟帮助科学家和工程师测试各种实验配置和产品设计，而不需要进行昂贵的实验。现在好莱坞电影中的许多特效都是通过计算机模拟来实现的。NVIDIA采用了与主要研究人员共同开发的BFECC方法进行烟雾模拟，http://developer.download.nvidia.com/SDK/10/direct3d/Source/Smoke/doc/Smoke.wmv.However，的计算机模拟是一个有噪声的过程。噪声经常来自机器误差，以及模拟对象的不光滑，如冲击、边界拐角、分离体内不同流体或组织的界面等。如果没有特殊技术，由冲击引起的模拟噪声很容易破坏模拟结果。事实上，开发计算方法的两个基本挑战是减少模拟时间和由于模拟对象的非光滑而产生的噪声。这位研究人员和他的同事们研究了一种新的去噪方法，这种方法更容易用于复杂的几何图形，对模拟对象的依赖程度更低。冲击模拟的初步结果令人鼓舞。这一新想法可以适用于许多其他领域，并推动改进计算方法的发展。例如，它可以更容易地模拟复杂的飞机形状，激发更强大的技术来稳定多相流体、燃料电池等的模拟，并为基础物理更具经验性的模拟提供黑匣子去噪工具。

High resolution capturing schemes for solving conservation laws, e.g., the ENO scheme, smear discontinuities within a few mesh cells and achieve high accuracy where the solution is smooth. A series of works by Cockburn and Shu et al. on discontinuous Galerkin (DG) methods and local DG introduce many new techniques to the DG family and enable it to solve a broader class of equations including conservation laws. Still, the limiting technique for DG is not very mature and is considered to be one of the major open problems in scientific computing. The investigator and his colleagues propose the further study of a new limiting technique, the hierarchical reconstruction (HR). It is a general reconstruction procedure used as a limiter to remove spurious oscillations in the presence of shocks. The HR algorithm, motivated by the moment limiter of Biswas, Devine and Flaherty (1994), involves only a MUSCL, a second order ENO or other piecewise linear reconstructions in each stage of a multi-layer reconstruction process without characteristic decomposition. Therefore it is compact and easy to implement for arbitrary meshes. It does not truncate higher degree terms of a polynomial and actually uses the information from all degree terms. It has been proved that HR does not reduce the approximation order of a polynomial. Moreover, HR can be used for finite volume and central schemes as well without characteristic decomposition, which leads to a new finite volume approach. The investigator and his colleagues also propose the study of the local constant velocity version of the back and forth error compensation and correction method (BFECC) for velocity advections in multi-phase fluid simulation, BFECC for moving meshes and for interpolation between grids. Adapting HR to BFECC wherever necessary could significantly improve the robustness of BFECC for non-smooth solution.This project is on the study and development of new methods for using computers to simulate certain natural phenomena such as airflow passing a wing, shock waves propagating in a body, smokes etc. Computer simulations help scientists and engineers testing various experimental configurations and product designs without conducting costly experiments.Nowadays a lot of special effects in Hollywood movies are made by computer simulation. The BFECC method co-developed by the principal investigator has been used by NVIDIA for smoke simulation, http://developer.download.nvidia.com/SDK/10/direct3d/Source/Smoke/doc/Smoke.wmv.However, computer simulation is a noisy process. Noises constantly come from machine errors, and from the non-smoothness of simulated objects, such as shocks, corners of boundaries, interfaces separating different fluids or tissues in a body etc. Without special techniques, simulation noises caused by shocks can easily destroy a simulation result. In fact, two of the fundamental challenges for developing computational methods are to reduce simulation time and noises from the non-smoothness of simulated objects. The investigator and his colleagues study a new method for removing noise, which is easier to use for complex geometry and less dependent on simulated objects. Preliminary results for simulations of shocks are encouraging. The new idea could be adapted to many other areas and motivate the development of improved computational methods. For example, it could allow a complicated aircraft shape to be simulated more easily, motivate more robust techniques to stabilize simulations of multi-phase fluids, fuel cells etc and provide a black-box de-noising tool for simulations whose underlying physics are more empirical.