Abstract
The majority of the most common physical phenomena can be described using partial differential equations (PDEs). However, they are very often characterized by strong nonlinearities. Such features lead to the coexistence of multiple solutions studied by the bifurcation theory. Unfortunately, in practical scenarios, one has to exploit numerical methods to compute the solutions of systems of PDEs, even if the classical techniques are usually able to compute only a single solution for any value of a parameter when more branches exist. In this work, we implemented an elaborated deflated continuation method that relies on the spectral element method (SEM) and on the reduced basis (RB) one to efficiently compute bifurcation diagrams with more parameters and more bifurcation points. The deflated continuation method can be obtained combining the classical continuation method and the deflation one: the former is used to entirely track each known branch of the diagram, while the latter is exploited to discover the new ones. Finally, when more than one parameter is considered, the efficiency of the computation is ensured by the fact that the diagrams can be computed during the online phase while, during the offline one, one only has to compute one-dimensional diagrams. In this work, after a more detailed description of the method, we will show the results that can be obtained using it to compute a bifurcation diagram associated with a problem governed by the Navier-Stokes equations.
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Acknowledgments
We acknowledge the support by European Union Funding for Research and Innovation - Horizon 2020 Program - in the framework of European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” (P.I. Prof. Gianluigi Rozza).
We also acknowledge the INDAM-GNCS project “Advanced intrusive and non-intrusive model order reduction techniques and applications”, 2019.
Funding
Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati - SISSA within the CRUI-CARE Agreement. This work is supported by European Union Funding for Research and Innovation - Horizon 2020 Program - European Research Council Executive Agency: Consolidator Grant H2020 ERC CoG 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics” (P.I. Prof. Gianluigi Rozza) and the INDAM-GNCS project “Advanced intrusive and non-intrusive model order reduction techniques and applications,” 2019.
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Communicated by: Stefan Volkwein
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Authors’ contributions
M. Pintore: running tests and drafting manuscript; F. Pichi: drafting manuscript and test assessment; M. Hess: general code and drafting manuscript; G. Rozza: supervision and preparation of activities; C. Canuto: supervision and manuscript polishing.
Data availability
The program used to obtain all the numerical results is ITHACA-SEM (https://mathlab.sissa.it/ithaca-sem).
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The code is available at https://github.com/mathLab/ITHACA-SEM.
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Pintore, M., Pichi, F., Hess, M. et al. Efficient computation of bifurcation diagrams with a deflated approach to reduced basis spectral element method. Adv Comput Math 47, 1 (2021). https://doi.org/10.1007/s10444-020-09827-6
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DOI: https://doi.org/10.1007/s10444-020-09827-6
Keywords
- Spectral element method
- Reduced basis method
- Reduced order model
- Deflated continuation method
- Bifurcation diagram
- Steady bifurcations
Mathematics Subject Classification (2010)
- 35B60
- 35Q35
- 37M20
- 76E30
- 76M22