Entropy stability for a fourth-order accurate finite-volume method for Burgers' equation
Date
2019
Authors
Meisner, Noah, author
Gao, Xinfeng, advisor
Guzik, Stephen, committee member
Liu, Jiannguo, committee member
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Abstract
Computational fluid dynamics (CFD) algorithms need efficiency, accuracy, and robustness to be useful to engineers. Faster computers improve the effective speed of a given method, and larger memories allow higher grid resolution, improving accuracy. However, robustness cannot be achieved through advancements in computer hardware. Improvements in this area require a fundamental understanding of the mathematical and physical aspects of the algorithm being investigated. For high-order numerical algorithms, the stability can easily be aggravated with the presence of strong gradients. Many methods in CFD incorporate some kind of numerical limiter to suppress spurious oscillations and handle nonlinear instabilities for flows with strong discontinuities. However, these limiters often lack a basis in the physics that governs the fluid flow. For this reason, the present research employs a limiting method that is based on the second law of thermodynamics to achieve numerical robustness for a higher order code in solving flows with strong discontinuities. The aim of this work is to address the question of robustness for a high-order finite-volume method (FVM) by extending the entropy stability strategy developed by Marshal L. Merriam for a second-order FVM. Unlike generic limiters or artificial viscosity, the approach explored in this thesis provides a physical, quantitative explanation for artificial viscosity or limiters in the form of entropy. The mathematical derivation of the entropy stability method is presented in detail, shortcomings of the method by Merriam are explored, and a more robust approach to deriving an entropy stable limiting method was carried out for the low-order methods. As a first step, this study focuses on the application to Burgers' equation for both a first- and second-order accurate solution to a problem with the onset of shocks. Then, a cell entropy fix for the fourth-order discretization scheme is derived and applied to Burgers' equations. Although the oscillations near the discontinuities can be mitigated, the logical conditions associated with ensuring the entropy constraints become impractical to implement for high-order discretization schemes. Through this research, it is deemed that the entropy stability method proposed by Merriam may not be a viable solution to effectively suppress oscillations near strong discontinuities of problems governed by systems of nonlinear equations, particularly, for high-order schemes.
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Subject
computational
finite-volume
algorithm
fluid
dynamics