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

翻译：Runge-Kutta 放松方法复制了非线性保护法的微粒稳定半分解的酶分解(或保存),在本文中,我们从放松龙格-Kutta 计划分离出来,这些计划适用于对最佳控制问题采取离散-最佳优化办法。此外,我们证明,衍生的离散放松龙格-库塔 adituto 保持了时间对称,用于ODE的线性Skew-对称系统。数字实验核查这些理论结果,同时表明在计算离散连接时适当处理放松参数的重要性。