An Eulerian-Lagrangian method for optimization problems governed by multidimensional nonlinear hyperbolic PDEs. (English) Zbl 1306.49047
Summary: 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.
MSC:
| 49M30 | Other numerical methods in calculus of variations (MSC2010) |
| 49M25 | Discrete approximations in optimal control |
| 49M05 | Numerical methods based on necessary conditions |
| 49K20 | Optimality conditions for problems involving partial differential equations |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 35L60 | First-order nonlinear hyperbolic equations |
| 65M08 | Finite volume methods for initial value and initial-boundary value problems involving PDEs |
Keywords:
optimal control; nonlinear hyperbolic partial differential equations; Eulerian finite-volume central-upwind scheme; Lagrangian discrete characteristics methodReferences:
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