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On the stability and accuracy of hig...

Reyna, Matthew.

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On the stability and accuracy of high-order Runge-Kutta discontinuous Galerkin methods.

紀錄類型:	書目-電子資源 : Monograph/item
正題名/作者:	On the stability and accuracy of high-order Runge-Kutta discontinuous Galerkin methods./
作者:	Reyna, Matthew.
出版者:	Ann Arbor : ProQuest Dissertations & Theses, : 2014,
面頁冊數:	105 p.
附註:	Source: Dissertations Abstracts International, Volume: 76-05, Section: B.
Contained By:	Dissertations Abstracts International76-05B.
標題:	Applied Mathematics. -
電子資源:	http://pqdd.sinica.edu.tw/twdaoapp/servlet/advanced?query=3643547
ISBN:	9781321312690

On the stability and accuracy of high-order Runge-Kutta discontinuous Galerkin methods.
Reyna, Matthew.

On the stability and accuracy of high-order Runge-Kutta discontinuous Galerkin methods. - Ann Arbor : ProQuest Dissertations & Theses, 2014 - 105 p.

Source: Dissertations Abstracts International, Volume: 76-05, Section: B.

Thesis (Ph.D.)--Rensselaer Polytechnic Institute, 2014.

This item must not be sold to any third party vendors.

Scientific, mathematical, and computational advances have made high-order numerical methods for hyperbolic conservation laws both increasingly important and increasingly accessible. However, various issues with the stability and accuracy of high-order methods can limit the appeal of these schemes. In this thesis, we consider such issues in the context of Runge-Kutta discontinuous Galerkin (RKDG) methods, which combine discontinuous Galerkin (DG) spatial discretizations with Runge-Kutta (RK) temporal discretizations. RKDG methods possess a number of appealing features for the numerical solution of hyperbolic problems, including compactness, hp-adaptivity, high parallelizability, and high-order accuracy. When combined with an RK method, high-order DG methods suffer from relatively restrictive stable time-step conditions compared to comparable finite difference methods, but the related high-order central DG methods generally enjoy larger time steps. In the first part of the thesis, we investigate the influence of the accuracy order of the DG and central DG methods on the stability of the fully discrete schemes. To understand this relationship mathematically, we derive analytical bounds on the DG and central DG spatial operators for the linear advection equation, and we obtain sufficient stability conditions for DG and central DG methods in space combined with locally stable RK methods in time. We validate our results numerically and extend them for discretizations of related problems. Moreover, the numerical treatment of inflow boundary conditions in high-order RKDG methods can affect the accuracy of the schemes, reducing them to lower order. In the second part of the thesis, we investigate various strategies for the treatment of the boundary conditions to understand their roles in the accuracy order of the overall RKDG schemes. We focus on the third-order and fourth-order RKDG discretizations of one-dimensional scalar and systems of linear and nonlinear conservation laws, providing analytical results in some cases and numerical results for more general problems. We especially explore the details of implementing the correction strategies for systems.

ISBN: 9781321312690Subjects--Topical Terms:

1669109
Applied Mathematics.
Subjects--Index Terms:

Accuracy

On the stability and accuracy of high-order Runge-Kutta discontinuous Galerkin methods.
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520 $a Scientific, mathematical, and computational advances have made high-order numerical methods for hyperbolic conservation laws both increasingly important and increasingly accessible. However, various issues with the stability and accuracy of high-order methods can limit the appeal of these schemes. In this thesis, we consider such issues in the context of Runge-Kutta discontinuous Galerkin (RKDG) methods, which combine discontinuous Galerkin (DG) spatial discretizations with Runge-Kutta (RK) temporal discretizations. RKDG methods possess a number of appealing features for the numerical solution of hyperbolic problems, including compactness, hp-adaptivity, high parallelizability, and high-order accuracy. When combined with an RK method, high-order DG methods suffer from relatively restrictive stable time-step conditions compared to comparable finite difference methods, but the related high-order central DG methods generally enjoy larger time steps. In the first part of the thesis, we investigate the influence of the accuracy order of the DG and central DG methods on the stability of the fully discrete schemes. To understand this relationship mathematically, we derive analytical bounds on the DG and central DG spatial operators for the linear advection equation, and we obtain sufficient stability conditions for DG and central DG methods in space combined with locally stable RK methods in time. We validate our results numerically and extend them for discretizations of related problems. Moreover, the numerical treatment of inflow boundary conditions in high-order RKDG methods can affect the accuracy of the schemes, reducing them to lower order. In the second part of the thesis, we investigate various strategies for the treatment of the boundary conditions to understand their roles in the accuracy order of the overall RKDG schemes. We focus on the third-order and fourth-order RKDG discretizations of one-dimensional scalar and systems of linear and nonlinear conservation laws, providing analytical results in some cases and numerical results for more general problems. We especially explore the details of implementing the correction strategies for systems.
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