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On the Consistency of Runge–Kutta Methods Up to Order Three Applied to the Optimal Control of Scalar Conservation Laws

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Numerical Analysis and Optimization (NAO 2017)

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Abstract

Higher-order Runge–Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK coefficients are derived that coincide with those characterizing strong stability preserving Runge–Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers’ equation validate the theoretical results.

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Acknowledgements

This work is supported by the German Research Foundation (DFG) within project B02 of CRC TRR 154.

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Authors and Affiliations

  1. Weierstrass Institute, Mohrenstr. 39, 10117, Berlin, Germany

    Michael Hintermüller & Nikolai Strogies

  2. Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Michael Hintermüller

Authors
  1. Michael Hintermüller
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  2. Nikolai Strogies
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Correspondence to Michael Hintermüller .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

    Mehiddin Al-Baali

  2. Dipartimento di Ingegneria Informatica, Modellistica, Elettronica e Sistemistica, Calabria University, Arcavacata, Italy

    Lucio Grandinetti

  3. Department Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

    Anton Purnama

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Hintermüller, M., Strogies, N. (2018). On the Consistency of Runge–Kutta Methods Up to Order Three Applied to the Optimal Control of Scalar Conservation Laws. In: Al-Baali, M., Grandinetti, L., Purnama, A. (eds) Numerical Analysis and Optimization. NAO 2017. Springer Proceedings in Mathematics & Statistics, vol 235. Springer, Cham. https://doi.org/10.1007/978-3-319-90026-1_6

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