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A kinetic scheme for the Navier-Stokes equations and high-order methods for hyperbolic conservation laws.

机译：Navier-Stokes方程的动力学方案和双曲守恒定律的高阶方法。

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摘要

This dissertation revolves around algorithm development in the context of numerical methods for hyperbolic conservation laws and the compressible Navier-Stokes equations, with particular emphasis on unstructured meshes. Three distinct topics may be identified: Firstly, a new kinetic scheme for the compressible Navier-Stokes equations is developed. Kinetic numerical schemes are based on the discretization of a probability density function. In the context of fluid flow such schemes have a natural basis rooted in the kinetic theory of gases. A significant advantage of kinetic schemes is that they allow a compact, completely mesh-independent discretization of the Navier-Stokes equations, which makes them well suited for next-generation solvers on general unstructured meshes. The new kinetic scheme is based on the Xu-Prenderaast BGK scheme, and achieves a dramatic reduction in computational cost, while also improving and clarifying the formulation with respect to the underlying kinetic gas theory.; The second topic addresses high-order numerical methods for conservation laws on unstructured meshes. High-order methods potentially produce higher accuracy with fewer degrees of freedom, compared to standard first or second order accurate schemes, while formulation for unstructured meshes makes complex computational domains amenable. The Spectral Difference Method offers a remarkably simple alternative to such high-order schemes for unstructured meshes as the Discontinuous Galerkin and Spectral Volume Methods. Significant contributions to the development of the Spectral Difference Method are presented, including stability analysis, viscous formulation, and h/p-multigrid convergence acceleration.; Finally, the theory of Gibbs-complementary reconstruction is utilized in the context of high-order numerical methods for hyperbolic equations. Gibbs-complementary reconstruction makes it possible to extract pointwise high-order convergence in the spectral approximation of non-smooth functions, despite the presence of the Gibbs phenomenon. Information is extracted from the spectral coefficients of the solution by reprojection onto a functional space endowed with certain properties, called a Gibbs-complementary space. This dissertation includes a proof of concept validating the technique on discontinuous solutions to nonlinear hyperbolic PDE, such as the Burgers equation and the Euler equations.
机译：本文围绕双曲守恒律和可压缩的Navier-Stokes方程的数值方法，特别是针对非结构化网格的算法开发进行了研究。可以确定三个不同的主题：首先，为可压缩的Navier-Stokes方程开发了一个新的动力学方案。动力学数值方案基于概率密度函数的离散化。在流体流动的情况下，这些方案具有扎根于气体动力学理论的自然基础。动力学方案的一个显着优势是，它们允许对Navier-Stokes方程进行紧凑，完全独立于网格的离散化，这使其非常适合于一般非结构化网格上的下一代求解器。新的动力学方案是基于Xu-Prenderaast BGK方案的，并大大降低了计算成本，同时还根据基本的动力学气体理论改进和阐明了公式。第二个主题涉及非结构网格上守恒律的高阶数值方法。与标准的一阶或二阶精度方案相比，高阶方法有可能以较少的自由度产生更高的精度，而对非结构化网格的公式化使得复杂的计算域可得到满足。对于非结构化网格，这种不连续Galerkin和光谱体积法可以很简单地替代光谱差异法。提出了对谱差法发展的重要贡献，包括稳定性分析，粘性公式化和h / p-multigrid收敛加速。最后，在双曲方程的高阶数值方法的背景下，采用了吉布斯互补重构理论。尽管存在吉布斯现象，但吉布斯互补重建使得有可能在非平滑函数的光谱近似中提取逐点高阶收敛。通过重新投影到具有某些特性的功能空间（称为吉布斯互补空间）上，从解决方案的光谱系数中提取信息。本论文包括一个概念证明，验证了非线性双曲型PDE不连续解的技术，例如Burgers方程和Euler方程。

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  • 作者

    May, Georg.;

  • 作者单位

    Stanford University.;

  • 授予单位 Stanford University.;
  • 学科 Engineering Aerospace.
  • 学位 Ph.D.
  • 年度 2006
  • 页码 136 p.
  • 总页数 136
  • 原文格式 PDF
  • 正文语种 eng
  • 中图分类 航空、航天技术的研究与探索;
  • 关键词

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