具有子网格式间断分辨率的隐式间断谱元法构造

The discontinuous spectral element method has been generalized to be an efficient scheme on mixed grids in solving compressible flows. This project studies the implementation of implicit time integration and sub-cell shock capturing strategy on discontinuous spectral element method. The Newton-Krylov method in implicit time uses large memory and thus grid number is limited in the calculation. This project uses a low memory implicit time integration scheme. The key ingredient is the p-multigrid strategy, where the explicit third order Runge-Kutta method is used in the level with high order schemes and the Jacobian-free preconditioned GMRES method is used for the levels with p0 and p1 schemes. This project proposes a projection WENO limiter, which uses the WENO procedure for the weighted summation of the high order central polynomial and the linear candidate polynomial constructed through the L2 project way. This limiter can maintain the high order accuracy in the smooth regions, capture the weak shock in sub-cell resolution and suppress the numerical oscillations in the inner degrees of freedom near strong shocks. The projection WENO limiter is combined with the adaptive mesh refinement to capture the strong shock in high resolution. The project aims at calculating the complicated turbulence flow such as shock-vortex interaction and multi-species reacting flows on complex geometries with discontinuous spectral element methods.

间断谱元法已被推广成为混合网格下可压缩流动计算的一种高效方法。本项目研究间断谱元法隐式时间积分实施方案和对间断的子网格式无波动捕捉策略。针对Newton-Krylov方法在高阶方法中占用内存大以至难以使用大网格量计算问题，本项目使用一种低内存占用的隐式时间积分方法。在p-multgrid求解每层过程中，对高阶格式采用显式三阶Runge-Kutta方法推进，对p0和p1格式采用Jacobian-Free预处理GMRES求解。项目提出一种投影式WENO限制器，采用WENO方法加权高阶多项式与经过L2投影构造的线性多项式。该限制器能够保持光滑区域的高精度，对弱激波进行子网格式间断捕捉，同时有效抑制强激波单元内自由度引起的数值振荡。投影式WENO限制器与网格自适应加密结合进一步提高强激波在间断附近分辨率。项目目标是要实现间断谱元法在复杂外形下高效率计算激波旋涡作用和多组分气体反应等复杂湍流流动。

本项目旨在构造一类高效率的高精度间断谱元法并用于求解包含激波等结构的复杂流动。首先通过节点DG格式无积分化的实现，并与张量积上的DGSEM格式整合成混合网格下的高精度格式，实现了一个更加实用且高效的DG框架，大量数值例子也表明其与基于积分的方法具有类似的非线性稳定性。第二是构造了一类能作用在混合网格的p加权限制器。该限制器对候选单元的构造利用了当地单元的逐级投影多项式和邻单元最佳逼近到中心单元的线性多项式。权重计算公式根据候选单元的构造形式而优化，并充分利用当地坐标减少计算量，最后达到光滑区域上的高精度和间断区域的振荡抑制。第三是实现PnP0的多重网格完全无矩阵的隐式时间格式。对高阶格式采用显式时间格式推进，对低阶格式采用隐式时间推进。高阶到低阶的插值过程采用投影的方法，低阶求解后的修正值用于修改高阶求解点值分布。最后将形成的高阶格式应用各类亚音速、跨音速和超音速可压缩流动计算，以及其它方程如中子通量的计算。

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