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An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs

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  • Alina Chertock
  • Michael Herty
  • Alexander Kurganov

Abstract

We present a numerical method for solving tracking-type optimal control problems subject to scalar nonlinear hyperbolic balance laws in one and two space dimensions. Our approach is based on the formal optimality system and requires numerical solutions of the hyperbolic balance law forward in time and its nonconservative adjoint equation backward in time. To this end, we develop a hybrid method, which utilizes advantages of both the Eulerian finite-volume central-upwind scheme (for solving the balance law) and the Lagrangian discrete characteristics method (for solving the adjoint transport equation). Experimental convergence rates as well as numerical results for optimization problems with both linear and nonlinear constraints and a duct design problem are presented. Copyright Springer Science+Business Media New York 2014

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:59:y:2014:i:3:p:689-724
    DOI: 10.1007/s10589-014-9655-y
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    References listed on IDEAS

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    1. E. M. Cliff & M. Heinkenschloss & A. R. Shenoy, 1997. "An Optimal Control Problem for Flows with Discontinuities," Journal of Optimization Theory and Applications, Springer, vol. 94(2), pages 273-309, August.
    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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    Cited by:

    1. 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.
    2. 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.

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