Moving Mesh Methods for Numerical Solution of Time Dependent Partial Differential Equations in Two and Three Spatial Dimensions

9626107黄等人发展了一种求解二维和三维含时偏微分方程组的移动网格方法，该方法在一维空间中推广了由研究人员和他的同事提出的移动网格偏微分方程组方法。用这种方法，基于热流方程和调和映射理论，显式地建立了MMPDE。MMPDE用于移动网格，将节点集中在物理解变化较大的区域，并控制某些网格质量，特别是偏斜度。研究了对由底层物理偏微分方程和MMPDE组成的扩展系统进行有效离散化和求解的技术。文中考虑了许多相关问题--例如，任意连通区域参考网格的计算、流向控制的使用、预处理技术的研究、时空有限元方法的检验以及该方法的稳健性和可靠性的研究。此外，该方法还被应用于翼型分析和机翼设计问题、移动边界问题以及涉及爆破解的问题。新的算法和计算技术同时在数学软件中实现，这可能会在科学家和工程师中得到广泛应用。该项目涉及开发新的计算方法，这些方法对于提高科学家和工程师解决大规模计算问题的能力至关重要，这些问题对我们的经济、环境和安全至关重要。研究的重点是开发所谓的自适应数值技术，以适应正在解决的特定问题的特殊特征。更具体地说，逼近科学问题解的地方(网格点)随时间移动以适应解的变化。网格自适应最近在问题的数值解中发挥了不可或缺的作用，因为尽管有超级计算机，这样的问题通常不能以其他方式得到令人满意的解决。这是因为在许多科学和工程问题中，只有一小部分物理域的解在网格中非常小的间隔内发生了巨大的变化。使用固定的均匀网格对这些问题进行数值求解是非常困难的，因为需要数百万个网格点来解决物理现象。另一方面，使用自适应网格方法可以显著减少网格点的数量，从而获得经济。与其他自适应网格方法不同，所研究的移动网格方法能够及时平滑地改变网格，以适应关键的解特征。它适合于并行计算。在许多工业制造问题的数值模拟中，动网格方法应该是非常有用的。一个特别的领域是翼型分析和机翼设计。例如，在翼型和机翼设计中，当指定压力分布时，将计算机翼或机翼形状。如果能像这些方法那样同时更新流动解和计算网格，从而更新翼型形状，将极大地节省计算资源。这一优势也适用于其他各种应用，例如研究工业和房屋火灾是如何以及为什么开始的问题。研究人员认为，“移动网格”方法是减少计算时间和提高精度的最佳方法，这些方法的实际技术转让近在咫尺。

9626107 Huang The investigator develops a moving mesh method for the numerical solution of two- and three-dimensional time-dependent PDEs (partial differential equations) that extends the MMPDE (Moving Mesh PDE) approach introduced by the investigator and his coworkers for PDEs in one spatial dimension. With this approach, an MMPDE is formulated explicitly based upon the heat flow equation and the theory of harmonic maps. The MMPDE is employed to move the mesh, to concentrate nodes in regions where the physical solution has large variations, and to control certain mesh qualities, in particular the skewness. Techniques are investigated to discretize and solve efficiently the extended system consisting of the underlying physical PDE and the MMPDE. A number of related issues are considered - e.g., the computation of a reference mesh for an arbitrary connected domain, the use of flow directional control, the study of preconditioning techniques, the examination of the space-time finite element method, and the investigation of robustness and reliability of the method. Moreover, the method is applied to problems in airfoil analysis and wing design, moving boundary problems, and problems involving blow-up solutions. The new algorithms and computational techniques are simultaneously implemented in mathematical software that could find widespread usage among scientists and engineers. This project is concerned with the development of new computational methods that are essential to enhance the ability of scientists and engineers to solve large scale computational problems that are crucial to our economy, environment, and security. The research is focused on development of so-called adaptive numerical techniques, where the special features of the particular problem being solved are adapted to. More specifically, the places (mesh points) where the solution to the scientific problem is being approximated are moved with time to adapt to the changes in the solu tion. Mesh adaptation has recently played an indispensable role in the numerical solution of the problems, as, supercomputers notwithstanding, such problems can generally not otherwise be solved satisfactorily. This is because in many problems of science and engineering, there is a small portion of the physical domain where large changes in the solution occur over very small separations in the mesh. Numerical solution of these problems using fixed uniform meshes is formidable because millions of mesh points are required to resolve the physical phenomena. On the other hand, use of adaptive mesh methods can significantly reduce the number of mesh points and thus economies can be gained. Unlike other adaptive mesh methods, the moving mesh method under study changes the mesh smoothly in time to adapt to the key solution features. It is suitable for parallel computing. The moving mesh method should be very useful in the numerical simulation of many industrial manufacturing problems. A particular area is airfoil analysis and wing design. For example, in the airfoil and wing design the airfoil or wing shape is computed when a pressure distribution is specified. It will save computing resources tremendously if the flow solution and the computational mesh, and hence the airfoil shape, can be updated simultaneously, as these methods do. This advantage holds for a variety of other applications as well, such as the problem of studying how and why industrial and house fires start. The investigator believes that a `moving mesh' approach is the best way to reduce computing time and improve accuracy, and that the actual technology transfer for these methods is close at hand.