Navier-Stokes方程自适应非结构网格虚拟单元浸入边界保正性间断有限元方法研究

We face three problems in the study of the Navier-Stokes equations: the generation of the computing grids, the reconstruction of the high order accuracy schemes and the improvement of the computational efficiency. By screening and analyzing the existing immersed boundary methods, we try to get a suitable engineering one for the purpose of reducing the difficulty of the grid generation and making the numerical simulation easier. As to the computing schemes, we try to introduce a new spatial compact WENO limiter which has positivity-preserving property for the positivity-preserving discontinuous Galerkin (DG) method. By using such property, we can get the correct density and pressure and can obtain good numerical results over the complex body boundaries and adaptive regions. The spatial stencils for the needs of reconstructing the WENO scheme are limited in the Von Neumann neighborhood and this procedure is easily extended to the adaptive meshes. By doing so, we can loosen the difficulty in dealing with the immersed boundaries, increase the numerical accuracy of the calculations and improve the efficiency of the adaptive algorithm for the purpose of the engineering application requirements.

在进行Navier-Stokes方程研究时，我们面临计算网格生成、高精度计算格式构造以及提高计算运行效率这三方面的问题。通过对已有浸入边界方法筛选和分析，得到较适合工程计算的方法，降低网格生成难度，使数值模拟对网格的要求降低；在计算格式方面，我们试图将一种空间模板紧凑的新型保正性WENO限制器引进保正性间断有限元方法中去。利用保正性特性得到正确的密度和压强值，能在复杂物体边界和自适应网格区域等处进行正确的数值模拟；将WENO重构的模板局限在Von Neumann邻域中,易于推广到自适应情形。能有效降低虚拟单元方法在浸入边界处的处理难度，提高计算精度，改善自适应算法的执行效率，达到工程应用要求。

我们在相关研究工作中能高效生成计算网格、构造了多个具有广泛工程应用背景的高精度计算格式。通过对已有浸入边界方法筛选和分析，得到较适合工程计算的方法，能有效降低网格生成难度，使数值模拟对网格的要求降低；在计算格式方面，我们构造了空间模板紧凑的新型保正性HWENO计算格式和多个WENO限制器引进间断有限元方法中去。能利用保正性特性得到正确的密度和压强值，能在复杂物体边界和自适应网格区域等处进行正确的数值模拟；将WENO 重构的模板局限在Von Neumann邻域中,易于推广到自适应情形。能有效降低虚拟单元方法在浸入边界处的处理难度，提高计算精度，改善自适应算法的执行效率，达到工程应用要求。