High-order energy stable flux reconstruction schemes for fluid flow simulations on unstructured grids

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Nowadays, most commercial CFD software relies exclusively on low-order methods (methods for which the spatial order of accuracy is at most two) for the simulation of flows over complex geometries. Although these methods are extremely robust and efficient, they are often inadequate when simulations require a very high level of accuracy. As desired error tolerances continue to decrease, high-order methods for unstructured grids will continue to grow in popularity. In the context of fluid flow simulations, applications that require high levels of accuracy (which could therefore benefit greatly from the use of high-order methods) include Direct Numerical Simulations (DNS), Large-Eddy Simulations (LES), Computational Aero-Acoustics (CAA) and vortex dominated flows. In 2007, Huynh presented the Flux Reconstruction (FR) approach to high-order methods. The method is attractive because it is intuitive, straightforward to implement, unifying (in the sense that it can recover various well known high-order methods), computationally efficient (since it does not require the use of numerical integration), and easily parallelizable. However, there remained questions regarding the stability of FR schemes and their applicability to practical fluid flow problems. In this thesis, a new class of FR schemes is identified, and applied to solve the 3D Navier-Stokes equations on mixed unstructured grids. The new schemes are referred to as Energy Stable Flux Reconstruction (ESFR) schemes. Their implementation in 1D, on triangular and quadrilateral elements in 2D, and on prismatic and hexahedral elements in 3D is discussed. The stability and accuracy properties of ESFR schemes are studied thoroughly. An energy stability proof is used to show the linear stability of the schemes for all orders of accuracy, in 1D and on triangular elements in 2D. Various numerical experiments in the field of fluid dynamics are then used to demonstrate their effectiveness. The implementation of ESFR schemes on Graphics Processing Units (GPUs) is also discussed. Because of their high arithmetic intensity and their element-local nature, the schemes are well suited for new massively parallel hardware architectures such as GPUs. The efficient implementation of ESFR schemes on GPUs results in speedups of at least one order of magnitude relative to traditional high-order methods for unstructured grids running on conventional computer hardware.


Type of resource text
Form electronic; electronic resource; remote
Extent 1 online resource.
Publication date 2012
Issuance monographic
Language English


Associated with Castonguay, Patrice
Associated with Stanford University, Department of Aeronautics and Astronautics
Primary advisor Jameson, Antony, 1934-
Thesis advisor Jameson, Antony, 1934-
Thesis advisor Lele, Sanjiva K. (Sanjiva Keshava), 1958-
Thesis advisor MacCormack, R. W. (Robert William), 1940-
Advisor Lele, Sanjiva K. (Sanjiva Keshava), 1958-
Advisor MacCormack, R. W. (Robert William), 1940-


Genre Theses

Bibliographic information

Statement of responsibility Patrice Castonguay.
Note Submitted to the Department of Aeronautics and Astronautics.
Thesis Thesis (Ph.D.)--Stanford University, 2012.
Location electronic resource

Access conditions

© 2012 by Patrice Castonguay
This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).

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