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

所提出的工作的主要思想在于使用离散形式的熵平衡方程作为控制有限元（FE）可压缩气体模拟中人工耗散的最佳量的基本原理。 熵控制可以根据所谓的修正熵函数重新解释为非线性稳定性估计。 初步结果包括使用泰勒-伽辽金时间离散方法与休斯和约翰逊提出的人工粘度模型相结合对所提出的想法进行了实际验证，所有这些都在 h 自适应线性有限元的背景下进行。获得的数值结果证实，熵控制确实可以为有限元离散化中的稳定性和高阶分辨率之间的仔细平衡提供基础。 拟议的研究包括：局部熵控制的研究（在迄今为止获得的结果中，仅保证了全局稳定性）、其他时间离散方法和人工耗散模型的研究以及对高阶空间近似的扩展。 拟议的工作涉及超音速粘性和无粘流，通过可压缩纳维-斯托克斯和欧拉方程进行建模。 可能的应用包括对整个飞机或其一部分周围的流动进行建模，以及主要与燃烧问题（发动机）相关的内部流动。 所述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，应将控制稳定性的内部机制构建到现有的有限元代码中，从而产生一类新的、可靠的和强大的近似值。

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.