Adaptive Wavelet Element Methods for Highly Parallel Computations

自适应小波元方法是一种新的数学方法，从根本上比现有方法更适合于高度并行计算。本项目研究基于自适应小波方法的可扩展并行算法和模拟，用于研究燃烧流动问题。该项目是由工业应用的内在物理特性所要求的大规模模拟能力的需求驱动的。本研究的重点是燃烧流动问题的并行小波模拟，包括相关的算法/数学开发和软件实现。在小波分解和区域分解的基础上，提出了一种新的方法——小波元法。新方法将小波方法的适用性扩展到燃烧问题和其他更复杂几何形状的粘性流动问题，并提供了基于区域分解的粗粒并行性。它具有高效、稳定、高精度、高度并行性和良好的数据局部性。小波支持空间域和频域的定位，使自适应网格生成成为可能。小波系数唯一地表示了子域间工作量的变化，而小波元方法的并行性来自于两个层面——每个子域的样条小波ADI的并行分解水平和并行解水平。本文将主要完成以下四项工作：1)开发一种两级并行化小波元方法，该方法结合了域分解和小波ADI方法，以最大限度地提高并行性、局部性和可扩展性。2)研究小波元背景下元素（子域）之间的接口条件，研究其对求解精度、工作负载平衡和通信成本的影响。3)研究基于小波系数在域分解层次和各子域的ADI小波解的动态工作量平衡。4)对多维预混火焰的爆燃爆轰过渡问题进行了可扩展并行自适应小波元方法的测试。

The Adaptive Wavelet element method is a new mathematical method which is fundamentally more appropriate for highly parallel computing than existing methods. This project studies scalable parallel algorithms and simulations based on adaptive wavelet methods for the investigation of combustion flow problems. The project isa driven by the need for large scale simulation capability required by the intrinsic physical property of industrial applications.The focus of this research is on parallel wavelet simulation of combustion flow problems, which includes the associated algorithmic/mathematical development and software implementation. A new method - the Wavelet Element Method - is introduced based on wavelet and domain decomposition methods. The new method extends the applicability of wavelet methods to combustion problems and other viscous flow problems in more complex geometries and provides a coarse grain parallelism based on domain decomposition. It is efficient, stable, highly accurate, and highly parallel, and it has good data locality. Wavelets support localization in both space and frequency domain which makes adaptive grid generation possible. Wavelet coefficients uniquely indicate the workload variation across subdomains, and parallelism in the wavelet element method comes from two levels - the level of domain decomposition and the level of parallel solution of Spline Wavelet ADI in each subdomain.Four main tasks will be performed: 1) Develop a two level parallelized wavelet element method, which combines the domain decomposition and wavelet ADI methods, to maximize parallelism, locality, and scalability. 2) Investigate interface conditions among elements (subdomains) in the context of wavelet element and investigate their effects on solution accuracy, workload balance, and communication cost. 3) Study dynamic workload balance based on wavelet coefficients on both domain decomposition level and on ADI wavelet solution in each subdomain. 4) Test the scalable parallel adaptive wavelet element methods on the Deflagration to Detonation Transition Problems for multidimensional premixed flames.