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An adaptive finite element method for the compressible Euler equations
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
2009 (English)Licentiate thesis, comprehensive summary (Other academic)
Abstract [en]

This work develops a stabilized finite element method for the compressible Euler equations and proves an a posteriori error estimate for the approximated solution. The equations are approximated by the cG(1)cG(1) finite element method with continuous piecewise linear functions in space and time. cG(1)cG(1) gives a second order accuracy in space, and corresponds to a Crank-Nicholson type of discretization in time, resulting in second order accuracy in space, without a stabilization term.

The method is stabilized by componentwise weighted least squares stabilization of the convection terms, and residual based shock capturing. This choice of stabilization gives a symmetric stabilization matrix in the discrete system. The method is successfully implemented for a number of benchmark problems in 1D, 2D and 3D. We observe that cG(1)cG(1) with the above choice of stabilization is robust and converges to an accurate solution with residual based adaptive mesh refinement.

We then extend the General Galerkin framework from incompressible to compressible flow, with duality based a posteriori error estimation of some quantity of interest. The quantities of interest can be stresses, strains, drag and lift forces, surface forces or a mean value of some quantity. In this work we prove a duality based a posteriori error estimate for the compressible equations, as an extension of the earlier work for incompressible flow [25].

The implementation and analysis are validated in computational tests both with respect to the stabilization and the duality based adaptation

 

 

 

Place, publisher, year, edition, pages
Stockholm: KTH , 2009. , xii, 39 p.
Series
Trita-CSC-A, ISSN 1653-5723 ; 2009:13
Identifiers
URN: urn:nbn:se:kth:diva-10582ISBN: 978-91-7415-365-1 (print)OAI: oai:DiVA.org:kth-10582DiVA: diva2:219786
Presentation
2009-06-10, D42, KTH, Lindstedtsvägen 5, Plan 4, Stockholm, 14:15 (English)
Opponent
Supervisors
Available from: 2009-05-28 Created: 2009-05-28 Last updated: 2010-10-20Bibliographically approved
List of papers
1. A General Galerkin Finite Element Method for the Compressible Euler Equations
Open this publication in new window or tab >>A General Galerkin Finite Element Method for the Compressible Euler Equations
2008 (English)In: SIAM Journal on Scientific Computing, ISSN 1064-8275, E-ISSN 1095-7197Article in journal (Refereed) Submitted
Abstract [en]

In this paper we present a General Galerkin (G2) method for the compressible Euler equations, including turbulent ow. The G2 method presented in this paper is a nite element method with linear approximation in space and time, with componentwise stabilization in the form  of streamline diusion and shock-capturing modi cations. The method conserves mass, momentum  and energy, and we prove an a posteriori version of the 2nd Law of thermodynamics for the method.  We illustrate the method for a laminar shock tube problem for which there exists an exact analytical  solution, and also for a turbulent flow problem

Keyword
General Galerkin G2 method, stabilized finite element method, turbulent compressible flow, second law of thermodynamics
National Category
Mathematics
Identifiers
urn:nbn:se:kth:diva-25410 (URN)
Note
QS 20120314Available from: 2010-10-20 Created: 2010-10-20 Last updated: 2017-12-12Bibliographically approved
2. An adaptive finite element method for the compressible Euler equations
Open this publication in new window or tab >>An adaptive finite element method for the compressible Euler equations
2010 (English)In: INT J NUMER METHOD FLUID, 2010, Vol. 64, no 10-12, 1102-1128 p.Conference paper, Published paper (Refereed)
Abstract [en]

We present an adaptive finite element method for the compressible Euler equations, based on a posteriori error estimation of a quantity of interest in terms of a dual problem for the linearized equations. Continuous piecewise linear approximation is used in space and time, with componentwise weighted least-squares stabilization of convection terms and residual-based shock-capturing. The adaptive algorithm is demonstrated numerically for the quantity of interest being the drag force on a body.

Keyword
adaptive finite element method, a posteriori error estimation, dual problem, compressible euler equations, circular cylinder, wedge, sphere
National Category
Computer and Information Science
Identifiers
urn:nbn:se:kth:diva-25412 (URN)
Conference
15th International Conference on Finite Elements in Flow Problems Tokyo, JAPAN, APR 01-03, 2009
Note
QC 20101020Available from: 2010-10-20 Created: 2010-10-20 Last updated: 2012-01-16Bibliographically approved

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