Generalization of partitioned Runge–Kutta methods for adjoint systems

This study computes the gradient of a function numerical solutions ordinary differential equations (ODEs) with respect to initial condition. The adjoint method approximately by solving corresponding system numerically. In this context, Sanz-Serna [SIAM Rev., 58 (2016), pp. 3–33] showed that when value problem is solved Runge–Kutta (RK) method, can be exactly computed applying an appropriate RK system. Focusing on case where partitioned (PRK) paper presents which seen as generalization PRK methods, for gives exact gradient.