  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.
