Commutative properties for conservative space-time DG discretizations of optimal control problems involving the viscous Burgers equation

Abstract

We consider one-dimensional distributed optimal control problems with the state equa-tion being given by the viscous Burgers equation. We discretize using a space-time dis-continuous Galerkin approach. We use upwind flux in time and the symmetric interior penalty approach for discretizing the viscous term. Our focus is on the discretization of the convection terms. We aim for using conservative discretizations for the convection terms in both the state and the adjoint equation, while ensuring that the approaches of discretize-then-optimize and optimize-then-discretize commute. We show that this is possible if the arising source term in the adjoint equation is discretized properly, following the ideas of well-balanced discretizations for balance laws. We support our findings by numerical results.