  Symmetric method and symplectic method are classical notions in the theory of
Runge-Kutta methods. They can generate numerical flows that respectively
preserve the symmetry and symplecticity of the continuous flows in the phase
space. Adjoint method is an important way of constructing a new Runge-Kutta
method via the symmetrisation of another Runge-Kutta method. In this paper, we
introduce a new notion, called symplectic-adjoint Runge-Kutta method. We prove
some interesting properties of the symmetric-adjoint and symplectic-adjoint
methods. These properties reveal some intrinsic connections among several
classical classes of Runge-Kutta methods. In particular, the newly introduced
notion and the corresponding properties enable us to develop a novel and
practical approach of constructing high-order explicit Runge-Kutta methods,
which is a challenging and longly overlooked topic in the theory of Runge-Kutta
methods.
