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

Author

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  • 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
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    References listed on IDEAS

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    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.
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