Mathematical Sciences: Entropy-Controlled Adaptive Finite Element Simulations of Compressible Gas Flow

所提出的工作的主要思想包括在使用离散形式的熵平衡方程作为控制的最佳量的人工耗散有限元（FE）可压缩气体模拟的基本原理。 熵控制可以被重新解释为一个非线性的稳定性估计的所谓的修正熵函数。 初步结果包括实际验证所提出的想法，使用泰勒-伽辽金离散化的时间方法结合人工粘度模型提出的休斯和约翰逊，所有的上下文中的h-自适应线性有限元。所得数值结果证实，熵控制确实可以提供一个基础，在FE离散的稳定性和高阶分辨率之间的仔细平衡。 建议的调查包括：一个研究的局部熵控制（在迄今为止获得的结果，只有一个全球性的稳定性得到保证），其他时间离散化方法和人工耗散模型的研究，并扩展到高阶空间approxima。 所提出的工作涉及超声速粘性和无粘流动，可压缩的Navier-Stokes和Euler方程建模。 可能的应用包括整个飞机或其一部分周围的流动建模，以及主要与燃烧问题（发动机）相关的内部流动。 所描述的研究应有助于更好地理解熵函数在可压缩气体模拟的全局和局部稳定性中的基本作用。基于熵控制，一个内部的机制，控制稳定性应内置到现有的有限元代码，从而在一个新的，可靠的和强大的近似类。

The principal idea of the proposed work consists in using a discrete form of the entropy balance equation as a rationale for controlling the optimal amount of artificial dissipation in Finite Element (FE) compressible gas simulations. The entropy control can be reinterpreted as a nonlinear stability estimate in terms of the so-called modified entropy function. Preliminary results include a practical verification of the proposed ideas, using the Taylor-Galerkin discretization in time method combined with an artificial viscosity model proposed by Hughes and Johnson, all in the context of h-adaptive linear finite elements. The obtained numerical results confirm that the entropy control indeed may provide a basis for the careful balance between stability and higher-order resolution in FE discretizations. The proposed investigations include: a study of a local entropy control (in the results obtained so far, only a global stability was ensured), a study of other time discretization methods and artificial dissipation models, and extensions to higher order spatial approxima tions. The proposed work deals with the supersonic viscous and inviscid flows, modeled by compressible Navier-Stokes and Euler equations. Possible applications include modeling of flows around a complete aircraft or part of it, and internal flows related mostly to combustion problems (engines). The described research should contribute to a better understanding of the fundamental role of the entropy function in global and local stability of compressible gas simulations. Based on the entropy control, an internal mechanism for controlling stability should be built into the existing FE codes, resulting in a new, reliable and powerful class of approximations.