Discrete Adjoint Computations for Relaxation Runge–Kutta Methods

Relaxation Runge–Kutta methods reproduce a fully discrete dissipation (or conservation) of entropy for stable semi-discretizations nonlinear conservation laws. In this paper we derive the adjoint relaxation schemes, which are applicable to discretize-then-optimize approaches optimal control problems. Furthermore, prove that derived preserves time-symmetry when applied linear skew-symmetric systems ODEs. Numerical experiments verify these theoretical results while demonstrating importance appropriately treating parameter computing adjoint.