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Runge-Kutta methods for linear semi-explicit operator differential-algebraic equations


Authors: R. Altmann and C. Zimmer
Journal: Math. Comp. 87 (2018), 149-174
MSC (2010): Primary 65J10, 65L80, 65M12
DOI: https://doi.org/10.1090/mcom/3270
Published electronically: June 21, 2017
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Abstract: As a first step towards time-stepping schemes for constrained PDE systems, this paper presents convergence results for the temporal discretization of operator DAEs. We consider linear, semi-explicit systems which include e.g. the Stokes equations or applications with boundary control. To guarantee unique approximations, we restrict the analysis to algebraically stable Runge-Kutta methods for which the stability functions satisfy $ R(\infty )=0$. As expected from the theory of DAEs, the convergence properties of the single variables differ and depend strongly on the assumed smoothness of the data.


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Additional Information

R. Altmann
Affiliation: Institut für Mathematik MA4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Email: raltmann@math.tu-berlin.de

C. Zimmer
Affiliation: Institut für Mathematik MA4-5, Technische Universität Berlin, Straße des 17. Juni 136, 10623 Berlin, Germany
Email: zimmer@math.tu-berlin.de

DOI: https://doi.org/10.1090/mcom/3270
Keywords: Operator DAEs, PDAEs, Runge-Kutta methods, implicit Euler scheme, regularization
Received by editor(s): March 31, 2016
Published electronically: June 21, 2017
Additional Notes: The work of the first author was supported by the ERC Advanced Grant "Modeling, Simulation and Control of Multi-Physics Systems" MODSIMCONMP. The work of the second author was supported by the Einstein Foundation Berlin within the project “Model reduction for complex transport-dominated phenomena and reactive flows”.
Article copyright: © Copyright 2017 American Mathematical Society

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