一类边界层问题基于时空指数离散的高性能数值方法研究

The numerical solution of compressible boundary layer problems is one of hotspots and difficulties in the field of computational fluid dynamics, and it is also the critical problem of many practical engineering problems including the shape optimization of advanced aircrafts, numerical weather prediction and pollutant diffusion problems. The traditional numerical methods, which improve the accuracy by refining meshes in the boundary layer, need much computation cost and are not easy to obtain high accuracy. It is extremely desired to develop fast and accurate numerical methods for compressible boundary layer problems. In this project, we will develop fast and accurate numerical methods for compressible boundary layer problems by combining seamlessly stable exponential time differencing with high accurate spatial discretization with exponential function and polynomial hybrid basis. Rigorous numerical stability and error estimates will be investigated for the proposed novel numerical methods. Furthermore, the predictor-corrector non-overlapping domain decomposition method and GPU-based acceleration techniques will be employed to develop the parallel version of proposed methods on high performance computing platforms. Finally, the proposed methods will be employed to solve some typical compressible boundary layer problems.

可压缩边界层问题的高效求解是计算流体力学领域的研究热点和研究难点之一，也是先进飞行器优化设计、数值天气预报和空中污染物扩散等很多科学与工程计算问题求解的关键。传统的通过边界层网格加密的方法计算量较大且很难达到高精度和较好的数值稳定性，发展新的可压缩边界层问题的高性能数值方法已经成为求解这一类问题的迫切需求。本项目尝试将最近发展的以指数函数和多项式作为混合基底的高精度空间离散方法与时间方向稳定的指数时间差分方法无缝耦合，发展求解可压缩边界层问题的快速高精度时空指数离散方法并进行相应的稳定性分析和误差估计；进一步，结合预估校正非重叠区域分解方法和GPU计算等加速技术，发展能够在高性能计算平台上运行的求解可压缩边界层问题的高性能数值方法，探索出一条高效求解可压缩边界层问题的新途径。