On the consistency of Runge--Kutta methods up to order three applied to the optimal control of scalar conservation laws
- Hintermüller, Michael
- Strogies, Nikolai
2010 Mathematics Subject Classification
- 49J15 49J20 35L65 65L06 65M06 65M12
- optimal control, conservation laws, discretization methods, RK methods, TVD-RK
Higher-order Runge-Kutta (RK) time discretization methods for the optimal control of scalar conservation laws are analyzed and numerically tested. The hyperbolic nature of the state system introduces specific requirements on discretization schemes such that the discrete adjoint states associated with the control problem converge as well. Moreover, conditions on the RK-coefficients are derived that coincide with those characterizing strong stability preserving Runge-Kutta methods. As a consequence, the optimal order for the adjoint state is limited, e.g., to two even in the case where the conservation law is discretized by a third-order method. Finally, numerical tests for controlling Burgers equation validate the theoretical results.
- Numerical Analysis and Optimization, M. Al-Baali, L. Grandinetti, A. Purnama, eds., vol. 235 of Springer Proceedings in Mathematics & Statistics, Springer Nature Switzerland AG, Cham, 2019, pp. 119--154.