An analysis of stability of the flux reconstruction formulation with applications to shock capturing

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High-order methods in Computational Fluid Dynamics (CFD) have been growing in popularity due to their promise of increased computational efficiency and fidelity to flow physics. Amongst a plethora of methods proposed over the last few decades, Discontinuous Galerkin (DG) type (Finite Element Methods (FEM)) have drawn great attention due to their attractive accuracy and stability properties, facility for performing arbitrarily high order computations and the capability to handle complex unstructured geometries, among other features. The Flux Reconstruction (FR) approach to high-order methods is a flexible, robust and simple to implement framework that has proven to be a promising alternative to the traditional DG schemes on parallel architectures like Graphical Processing Units (GPUs) since it pairs exceptionally well with explicit time-stepping methods. While high order methods have already successfully displayed significant improvements over low order methods on certain fronts, one of the major reasons limiting their industry-wide adoption is their inferior robustness relative to low order methods. These high order schemes are prone to developing instabilities while solving nonlinear problems and the issue compounds with increasing order, thereby requiring a com- promise between accuracy and stability. Instabilities due to discontinuous solutions or shocks that develop in compressible flows and aliasing instabilties are two of the most important ones. This dissertation is divided into two major parts. In the first part, the stability of the FR framework for solving linear advection and advection-diffusion equations on tensor product elements has been investigated and the approach has been proven to be stable for both problems. In the second part, a robust and simple to implement shock capturing method which can be adopted in any unstructured high order Finite Element (FE)-type method has been developed. The proposed method does not sabotage the accuracy of the solution in smooth regions and shows great promise in our numerical simulations.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2016
Issuance monographic
Language English


Associated with Sheshadri, Abhishek
Associated with Stanford University, Department of Aeronautics and Astronautics.
Primary advisor Jameson, Antony, 1934-
Thesis advisor Jameson, Antony, 1934-
Thesis advisor Alonso, Juan José, 1968-
Thesis advisor Lele, Sanjiva K. (Sanjiva Keshava), 1958-
Advisor Alonso, Juan José, 1968-
Advisor Lele, Sanjiva K. (Sanjiva Keshava), 1958-


Genre Theses

Bibliographic information

Statement of responsibility Abhishek Sheshadri.
Note Submitted to the Department of Aeronautics and Astronautics.
Thesis Thesis (Ph.D.)--Stanford University, 2016.
Location electronic resource

Access conditions

© 2016 by Abhishek Sheshadri
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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