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.

翻译：本文提出了一类基于显式对角隐式龙格-库塔（ESDIRK）方法的隐式子步积分算法族。所提方法在每个子步中均达到三阶一致性，因此首个子步始终采用梯形法则。本文首次证明：当实现耗散控制与无条件稳定性同步时，所提出的$ s $子步隐式方法（$ s\le6 $）可达到$ s $阶精度。因此，本文基于三、四、五、六子步的$ s $子步方法，开发、分析并比较了四种成本最优的高阶隐式算法。每种高阶隐式算法共享相同的有效刚度矩阵以实现最优谱特性。与现有算法不同，所提出高阶方法不会因求解受迫振动而产生阶次降低。此外，新方法克服了作者此前算法需额外计算加速度才能获得精确解的缺陷。通过线性和非线性算例验证了四种新型高阶算法的数值性能与优越性。