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

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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.