Presented By: Aerospace Engineering
AE Dissertation Defense by J. Brad Maeng
Aerospace Engineering Prof. Phil Roe is the Dissertation Committee Chair
Title: On the Advective Component of Active Flux Schemes for Nonlinear Hyperbolic Conservation Laws
A new class of numerical methods called the Active Flux methods is investigated for nonlinear hyperbolic conservation laws. The Active Flux methods are designed specifically to address the aspect that most modern compressible flow methods fail to do; the multidimensionality aspect. The Active Flux methods address the shortcoming by employing a two stage update process. In the first stage, nonconservative form of the system is introduced to provide the flexibility to pursue distinct numerical approaches for flow processes with differing physics. Because each process is treated separately, the numerical method can be appropriately formed to reflect each type of physics and to provide maximal stability. The Active Flux method concludes with the conservation update to produce a third-order accurate scheme.
The advective schemes of Active Flux is founded on the characteristic tracing method, or semi-Lagrangian method, which has long been used for developing numerical methods for hyperbolic problems. The first known Active Flux method for advection problems, Scheme V by van Leer, is revisited as a part of the development of advection scheme. The details of Scheme V are examined closely, and new improvements have been made to develop a more general nonlinear multidimensional advection scheme.
A detailed study of the nonlinear system of equations is made possible by the pressureless Euler system, which is the advective component of the Euler system. It serves as a stepping stone for the Euler system, and all necessary details of nonlinear system are explored. Lastly, an extension to the Euler system is presented, in which a novel nonlinear operator splitting method is introduced to correctly blend the contributions of nonlinear advection and acoustic processes. The Active Flux methods, as a result, produce a maximally stable, third-order accurate method for the multidimensional Euler system.
Some guiding principles of limiting are presented. Because two types of flow feature are kept separate in the numerical approach, the limiting process must also be kept separate. Advective problems obeying natural bounding principles are treated differently from acoustic problems with no explicit bounding principles. Distinct limiting approaches are explored along with discussions.
Doctoral Committee
Chair: Prof. Philip L Roe
Cognate Member: Prof. Robert Krasny
Members: Assoc. Prof. Karthik Duraisamy, Assoc. Prof. Krzysztof Fidkowski
Publications
“New approaches to limiting.” Philip L Roe, Tyler Lung and Jungyeoul Maeng. In 22nd AIAA Computational Fluid Dynamics Conference, 2015
“Active Flux Schemes for Hyperbolic Conservation Laws – Advective Component” J. B. Maeng, P. L. Roe, T. Eymann, D. Fan, In preparation 2017.
A new class of numerical methods called the Active Flux methods is investigated for nonlinear hyperbolic conservation laws. The Active Flux methods are designed specifically to address the aspect that most modern compressible flow methods fail to do; the multidimensionality aspect. The Active Flux methods address the shortcoming by employing a two stage update process. In the first stage, nonconservative form of the system is introduced to provide the flexibility to pursue distinct numerical approaches for flow processes with differing physics. Because each process is treated separately, the numerical method can be appropriately formed to reflect each type of physics and to provide maximal stability. The Active Flux method concludes with the conservation update to produce a third-order accurate scheme.
The advective schemes of Active Flux is founded on the characteristic tracing method, or semi-Lagrangian method, which has long been used for developing numerical methods for hyperbolic problems. The first known Active Flux method for advection problems, Scheme V by van Leer, is revisited as a part of the development of advection scheme. The details of Scheme V are examined closely, and new improvements have been made to develop a more general nonlinear multidimensional advection scheme.
A detailed study of the nonlinear system of equations is made possible by the pressureless Euler system, which is the advective component of the Euler system. It serves as a stepping stone for the Euler system, and all necessary details of nonlinear system are explored. Lastly, an extension to the Euler system is presented, in which a novel nonlinear operator splitting method is introduced to correctly blend the contributions of nonlinear advection and acoustic processes. The Active Flux methods, as a result, produce a maximally stable, third-order accurate method for the multidimensional Euler system.
Some guiding principles of limiting are presented. Because two types of flow feature are kept separate in the numerical approach, the limiting process must also be kept separate. Advective problems obeying natural bounding principles are treated differently from acoustic problems with no explicit bounding principles. Distinct limiting approaches are explored along with discussions.
Doctoral Committee
Chair: Prof. Philip L Roe
Cognate Member: Prof. Robert Krasny
Members: Assoc. Prof. Karthik Duraisamy, Assoc. Prof. Krzysztof Fidkowski
Publications
“New approaches to limiting.” Philip L Roe, Tyler Lung and Jungyeoul Maeng. In 22nd AIAA Computational Fluid Dynamics Conference, 2015
“Active Flux Schemes for Hyperbolic Conservation Laws – Advective Component” J. B. Maeng, P. L. Roe, T. Eymann, D. Fan, In preparation 2017.
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