In this paper, we are interested in high-order algorithms for time discretization and focus upon the high-order implicit/explicit algorithm designs. Five high-order unconditionally stable implicit algorithms, derived by the time continuous Galerkin method, weighting parameter method, collocation method, differential quadrature method, and the modified time-weighted residual method, in first-/second-order transient systems are taken into consideration. The present overview and contributions encompass: (1) The pros and cons of various methodologies for the design of high-order algorithms are first demonstrated. Generally, p unknown variables leads to the optimized \((2p-1)\)th-order accurate algorithms with controllable numerical dissipation, and/or 2pth-order accurate algorithms without controllable numerical dissipation. (2) Although it is claimed that the TCG method can achieve 2pth-order accuracy with controllable numerical dissipation, it will be shown in this paper that the conclusion was arrived via an inconsistent analysis for the accuracy and the controllable numerical dissipation. (3) Given the rapid increase on the computational cost for high-order algorithms, the iterative predictor/multi-corrector technique is applied to show a novel design for the high-order explicit algorithms derived from the high-order implicit algorithms for the first-order transient systems. Coupled with the high-order Legendre SEM (likewise isogeometric analysis, DG methods, p-version FEM, etc., can be employed) for the spatial discretization, this newly proposed explicit numerical framework can achieve and preserve high-order accuracy in both space and time. In comparison to the famous explicit Runge-Kutta method, these newly designed explicit algorithms have better solution accuracy with comparable stability region.

在本文中，我们对用于时间离散化的高阶算法感兴趣，并将重点放在高阶隐式/显式算法设计上。考虑一阶/二阶瞬态系统中时间连续Galerkin方法，加权参数方法，搭配方法，微分正交方法和改进的时间加权残差方法的五种高阶无条件稳定隐式算法。本概述和贡献包括：（1）首先展示了用于设计高阶算法的各种方法的利弊。通常，p个未知变量会导致优化的\（（2p-1）\）阶精确算法，其数值耗散可控制，和/或2 p没有可控制的数值耗散的三阶精确算法。（2）尽管声称TCG方法可以在2 p阶精度上实现可控的数值耗散，但本文将通过对精度和可控数值耗散的不一致分析得出结论。（3）鉴于高阶算法的计算成本迅速增加，应用迭代预测器/多重校正器技术展示了一种从高阶隐式算法派生而来的高阶显式算法的新颖设计。阶瞬态系统。结合高阶Legendre SEM（同样是等几何分析，DG方法，p可以使用版本有限元法等进行空间离散化，这种新提出的显式数值框架可以在空间和时间上实现并保持高阶精度。与著名的显式Runge-Kutta方法相比，这些新设计的显式算法在可比较的稳定区域内具有更好的求解精度。