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Equation-free, effective computation for discrete systems: a time stepper based approach
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.
KTH, School of Computer Science and Communication (CSC), Numerical Analysis and Computer Science, NADA.ORCID iD: 0000-0002-6321-8619
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2005 (English)In: International Journal of Bifurcation and Chaos in Applied Sciences and Engineering, ISSN 0218-1274, Vol. 15, no 3, 975-996 p.Article in journal (Refereed) Published
Abstract [en]

We propose a computer-assisted approach to studying the effective continuum behavior of spatially discrete evolution equations. The advantage of the approach is that the "coarse model" (the continuum, effective equation) need not be explicitly constructed. The method only uses a time-integration code for the discrete problem and judicious choices of initial data and integration times; our bifurcation computations are based on the so-called Recursive Projection Method (RPM) with arc-length continuation [Shroff & Keller, 1993]. The technique is used to monitor features of the genuinely discrete problem such as the pinning of coherent structures and its results are compared to quasi-continuum approaches such as the ones based on Pade approximations.

Place, publisher, year, edition, pages
2005. Vol. 15, no 3, 975-996 p.
Keyword [en]
equation-free methods, homogenization, discrete problems, bifurcation, pinning condition
National Category
Computer Engineering
Identifiers
URN: urn:nbn:se:kth:diva-4905DOI: 10.1142/S0218127405012399ISI: 000229230400016Scopus ID: 2-s2.0-21144445075OAI: oai:DiVA.org:kth-4905DiVA: diva2:6777
Note
Workshop on Modeling and Computations in Dynamical Systems Budapest, HUNGARY, OCT, 2003 QC 20101015Available from: 2005-02-02 Created: 2005-02-02 Last updated: 2017-12-06Bibliographically approved
In thesis
1. Aspects of the recursive projection method applied to flow calculations
Open this publication in new window or tab >>Aspects of the recursive projection method applied to flow calculations
2005 (English)Doctoral thesis, comprehensive summary (Other scientific)
Abstract [en]

In this thesis, we have investigated the Recursive Projection Method, RPM, as an accelerator for computations of both steady and unsteady flows, and as a stabilizer in a bifurcation analysis.

The criterion of basis extraction is discussed. It can be interpreted as a tolerance for the accuracy of the eigenspace spanned by the identified basis, alternatively it can be viewed as a criterion when the approximative Krylov sequence becomes numerically rank deficient.

Steady state calculations were performed on two different turbulent test-cases; a 2D supersonic nozzle flow with the Spalart-Allmaras 1-equation model and a 2D sub-sonic airfoil simulation using the κ - ε model. RPM accelerated the test-cases with a factor between 2 and 5.

In multi-scale problems, it is often of interest to model the macro-scale behavior, still retaining the essential features of the full systems. The ``coarse time stepper'' is a heuristic approach for circumventing the analytical derivation of models. The system studied here is a linear lattice of non-linear reaction sites coupled by diffusion. After reformulation of the time-evolution equation as a fixed-point scheme, RPM coupled with arc-length continuation is used to calculate the bifurcation diagrams of the effective (but analytically unavailable) equation.

Within the frame-work of dual time-stepping, a common approach in unsteady CFD-simulation, RPM is used to accelerate the convergence. Two test-cases were investigated; the von Karman vortex-street behind a cylinder at Re=100, and the periodic shock oscillation of a symmetric airfoil at M ∞ = 0.76 with a Reynolds number Re=11 x 106.

It was believed that once a basis had been identified, it could be retained for several steps. The simulations usually showed that the basis could only be retained for one step.

The need for updating the basis motivates the use of Krylov methods. The most common method is the (Block-) Arnoldi algorithm. As the iteration proceeds, Krylov methods become increasingly expensive and restart is required. Two different restart algorithm were tested. The first is that of Lehoucq and Maschhoff, which uses a shifted QR iteration, the second is a block extension of the single-vector Arnoldi method due to Stewart. A flexible hybrid algorithm is derived combining the best features of the two.

Place, publisher, year, edition, pages
Stockholm: KTH, 2005. ix, 26 p.
Series
Trita-NA, ISSN 0348-2952 ; 0444
Keyword
Datorteknik, applied mechanics, computer science, aerospace, Datorteknik
National Category
Computer Engineering
Identifiers
urn:nbn:se:kth:diva-101 (URN)91-7283-940-6 (ISBN)
Public defence
2005-01-20, Sal L1, Drottning Kristinas väg 30, Stockholm, 10:15
Opponent
Supervisors
Note

QC 20101015

Available from: 2005-02-02 Created: 2005-02-02 Last updated: 2012-09-21Bibliographically approved

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Runborg, Olof

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