Adjoint IMEX-based schemes for control problems governed by hyperbolic conservation laws

Starting from relaxation schemes for hyperbolic conservation laws we derive continuous and discrete schemes for optimization problems subject to nonlinear, scalar hyperbolic conservation laws. We discuss properties of first- and second-order discrete schemes and show their relations to existing results. In particular, we introduce first and second-order relaxation and relaxed schemes for both adjoint and forward equations. We give numerical results including tracking type problems with non-smooth desired states.

[1]  Yann Brenier,et al.  On a relaxation approximation of the incompressible Navier-Stokes equations , 2003 .

[2]  R. Natalini,et al.  Convergence of relaxation schemes for conservation laws , 1996 .

[3]  F. James,et al.  One-dimensional transport equations with discontinuous coefficients , 1998 .

[4]  William W. Hager,et al.  Runge-Kutta methods in optimal control and the transformed adjoint system , 2000, Numerische Mathematik.

[5]  Stefan Ulbrich,et al.  Adjoint-based derivative computations for the optimal control of discontinuous solutions of hyperbolic conservation laws , 2003, Syst. Control. Lett..

[6]  R. Natalini.  Convergence to equilibrium for the relaxation approximations of conservation laws , 1996 .

[7]  Paul H. Calamai,et al.  Projected gradient methods for linearly constrained problems , 1987, Math. Program..

[8]  Carl Tim Kelley,et al.  Iterative methods for optimization , 1999, Frontiers in applied mathematics.

[9]  A. Bressan,et al.  A variational calculus for discontinuous solutions of systems of conservation laws , 1995 .

[10]  CONVERGENCE OF A RELAXATION APPROXIMATION TO A BOUNDARY VALUE PROBLEM FOR CONSERVATION LAWS , 2001 .

[11]  Adrian Sandu,et al.  On the properties of discrete adjoints of numerical methods for the advection equation , 2008 .

[12]  G. Russo,et al.  Implicit-explicit Runge-Kutta schemes for stiff systems of differential equations , 2000 .

[13]  F. James,et al.  Differentiability with Respect to Initial Data for a Scalar Conservation Law , 1999 .

[14]  Peter Spellucci,et al.  Numerische Verfahren der nichtlinearen Optimierung , 1993 .

[15]  Stefano Bianchini,et al.  ON THE SHIFT DIFFERENTIABILITY OF THE FLOW GENERATED BY A HYPERBOLIC SYSTEM OF CONSERVATION LAWS , 2000 .

[16]  Chi-Wang Shu,et al.  Strong Stability-Preserving High-Order Time Discretization Methods , 2001, SIAM Rev..

[17]  Mapundi K. Banda,et al.  Higher-order relaxation schemes for hyperbolic systems of conservation laws , 2005, J. Num. Math..

[18]  R. M. COLOMBO,et al.  Optimal Control in Networks of Pipes and Canals , 2009, SIAM J. Control. Optim..

[19]  Stefan Ulbrich,et al.  A Sensitivity and Adjoint Calculus for Discontinuous Solutions of Hyperbolic Conservation Laws with Source Terms , 2002, SIAM J. Control. Optim..

[20]  A. Bressan,et al.  Shift-differentiability of the flow generated by a conservation law , 1996 .

[21]  E. Tadmor,et al.  Hyperbolic Problems: Theory, Numerics, Applications , 2003 .

[22]  Irena Lasiecka,et al.  Control Methods in PDE-Dynamical Systems , 2007 .

[23]  M. Banda,et al.  Relaxation WENO schemes for multidimensional hyperbolic systems of conservation laws , 2007 .

[24]  Michael Herty,et al.  An Eulerian–Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs , 2014, Comput. Optim. Appl..

[25]  Christian A. Ringhofer,et al.  Control of Continuum Models of Production Systems , 2010, IEEE Transactions on Automatic Control.

[26]  B. François,et al.  Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness , 1999 .

[27]  Hebe de Azevedo Biagioni,et al.  A Nonlinear Theory of Generalized Functions , 1990 .

[28]  Z. Xin,et al.  The relaxation schemes for systems of conservation laws in arbitrary space dimensions , 1995 .

[29]  Christian A. Ringhofer,et al.  A Continuum Model for a Re-entrant Factory , 2006, Oper. Res..