New Ideas for the Penalty MethodXie, Xihui (2022) New Ideas for the Penalty Method. Doctoral Dissertation, University of Pittsburgh. (Unpublished) This is the latest version of this item.
AbstractMy research is directed at accurate predictions of the flow of fluids and what the fluid transports. This is essential for many critical engineering and scientific applications, including climate change and energy efficiency optimization. For example, 85\% of the energy in the US is generated by combustion, for which accurate simulation of turbulent mixing is critical for energy efficiency optimization. To address these, my research develops algorithms that have the potential to break current barriers in accuracy, reliability and efficiency in CFD, and a rigorous mathematical foundation addressing when they work and how they fail, at the crossroads of theory and practical computation. My research considers the adaptivity of the penalty parameter $\epsilon$ both in space and in time. In the first project, I consider the $\epsilon-$adaptive penalty methods for the Navier-Stokes equation. The unconditional stability is proven for velocity when adapting $\epsilon$. The stability of the velocity time derivative under conditions on the rate of change of the penalty parameter is also analyzed. The analysis and tests show that adapting $\epsilon$ in response to $\|\nabla\cdot u\|$ removes the problem of picking $\epsilon$ and yields good approximations for the velocity. The adaptive penalty parameter method is supplemented by also adapting the time-step. The penalty parameter $\epsilon$ and time-step are adapted independently. The second project proposes and analyzes a new adaptive penalty scheme, which picks the penalty parameter $\epsilon$ element by element small where $\|\nabla\cdot u\|$ is large. The research starts by analyzing and testing the new scheme in the most simple but interesting setting, the Stokes problem. Finally, this adaptive method is extended and tested on the incompressible Navier-Stokes equation on complex flow problems. The scheme is developed in the penalty method but also can be used to pick a grad-div stabilization parameter. Share
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