  This paper proposes an implicit family of sub-step integration algorithms
grounded in the explicit singly diagonally implicit Runge-Kutta (ESDIRK)
method. The proposed methods achieve third-order consistency per sub-step and
thus the trapezoidal rule is always employed in the first sub-step. This paper
demonstrates for the first time that the proposed $ s $-sub-step implicit
method with $ s\le6 $ can reach $ s $th-order accuracy when achieving
dissipation control and unconditional stability simultaneously. Hence, this
paper develops, analyzes, and compares four cost-optimal high-order implicit
algorithms within the present $ s $-sub-step method using three, four, five,
and six sub-steps. Each high-order implicit algorithm shares identical
effective stiffness matrices to achieve optimal spectral properties. Unlike the
published algorithms, the proposed high-order methods do not suffer from the
order reduction for solving forced vibrations. Moreover, the novel methods
overcome the defect that the authors' previous algorithms require an additional
solution to obtain accurate accelerations. Linear and nonlinear examples are
solved to confirm the numerical performance and superiority of four novel
high-order algorithms.
