IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v80y2021i1d10.1007_s10589-021-00295-2.html
   My bibliography  Save this article

A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws

Author

Listed:
  • David Frenzel

    (Technical University of Darmstadt)

  • Jens Lang

    (Technical University of Darmstadt)

Abstract

The weighted essentially non-oscillatory (WENO) methods are popular and effective spatial discretization methods for nonlinear hyperbolic partial differential equations. Although these methods are formally first-order accurate when a shock is present, they still have uniform high-order accuracy right up to the shock location. In this paper, we propose a novel third-order numerical method for solving optimal control problems subject to scalar nonlinear hyperbolic conservation laws. It is based on the first-disretize-then-optimize approach and combines a discrete adjoint WENO scheme of third order with the classical strong stability preserving three-stage third-order Runge–Kutta method SSPRK3. We analyze its approximation properties and apply it to optimal control problems of tracking-type with non-smooth target states. Comparisons to common first-order methods such as the Lax–Friedrichs and Engquist–Osher method show its great potential to achieve a higher accuracy along with good resolution around discontinuities.

Suggested Citation

  • David Frenzel & Jens Lang, 2021. "A third-order weighted essentially non-oscillatory scheme in optimal control problems governed by nonlinear hyperbolic conservation laws," Computational Optimization and Applications, Springer, vol. 80(1), pages 301-320, September.
  • Handle: RePEc:spr:coopap:v:80:y:2021:i:1:d:10.1007_s10589-021-00295-2
    DOI: 10.1007/s10589-021-00295-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-021-00295-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.

    File URL: https://libkey.io/10.1007/s10589-021-00295-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Alina Chertock & Michael Herty & Alexander Kurganov, 2014. "An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs," Computational Optimization and Applications, Springer, vol. 59(3), pages 689-724, December.
    2. Mapundi Banda & Michael Herty, 2012. "Adjoint IMEX-based schemes for control problems governed by hyperbolic conservation laws," Computational Optimization and Applications, Springer, vol. 51(2), pages 909-930, March.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Albi, G. & Herty, M. & Pareschi, L., 2019. "Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems," Applied Mathematics and Computation, Elsevier, vol. 354(C), pages 460-477.
    2. Jack Reilly & Samitha Samaranayake & Maria Laura Delle Monache & Walid Krichene & Paola Goatin & Alexandre M. Bayen, 2015. "Adjoint-Based Optimization on a Network of Discretized Scalar Conservation Laws with Applications to Coordinated Ramp Metering," Journal of Optimization Theory and Applications, Springer, vol. 167(2), pages 733-760, November.
    3. Alina Chertock & Michael Herty & Alexander Kurganov, 2014. "An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs," Computational Optimization and Applications, Springer, vol. 59(3), pages 689-724, December.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:coopap:v:80:y:2021:i:1:d:10.1007_s10589-021-00295-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.