In this paper, a novel stabilized time-weighted residual methodology under the umbrella of Petrov-Galerkin time finite element formulation is developed to design a generalized computational framework, which permits unconditionally stable, high-order time accuracy, and features with controllable numerical dissipation for solving transient first-order systems. Various unconditionally stable (A/L-stable) algorithms can be readily obtained in the proposed framework having not only high-order accuracy but also controllable numerical dissipation in the high frequency. Quadratic and cubic basis functions are utilized to illustrate the specific design process of the proposed method, which consequently ends up with numerous third-/fifth-order time accurate algorithms, ${\mathbf{QUAD}}_3(\gamma ,{\rho }^{\infty })$ and ${\mathbf{CUBE}}_5(\gamma ,{\rho }^{\infty })$, with controllable numerical dissipation. Comparing with the well-known implicit Runge-Kutta (RK) family of algorithms, these newly developed ${\mathbf{QUAD}}_3(\gamma ,{\rho }^{\infty })$ and ${\mathbf{CUBE}}_5(\gamma ,{\rho }^{\infty })$ can (a) recover the Radau IIA₃/RK3 and Radau IIA₅/RK5 schemes by certain selection of algorithmic parameters ($\gamma ,{\rho }^{\infty }$); (b) obtain numerous new algorithms with improved solution accuracy that is superior to the RK family of algorithms, and (c) has similar computational efficiency as that of the implicit RK family of algorithms. Several single/multi-degree of freedom (SDOF and MDOF) problems are investigated to validate the proposed developments. In addition, it is worth noting that the proposed methodology can also use high-order (not limited to the quadratic and cubic) basis functions to design more advanced schemes and can be integrated with high-order spatial discretiaztion methods in the solution of space-time PDEs, such as isogeometric methods, SEM, Discontinuous Galerkin (DGM), p-versions FEM, etc.

本文在Petrov-Galerkin时间有限元公式的框架下，开发了一种新颖的稳定时间加权残差方法，以设计通用计算框架，该框架允许无条件稳定，高阶时间精度以及具有可控数值耗散的特征解决瞬态一阶系统。在所提出的框架中，不仅具有高阶精度，而且具有可控制的高频数值耗散，可以轻松地获得各种无条件稳定（A / L稳定）算法。利用二次和三次基函数来说明所提出方法的具体设计过程，因此最终产生了许多三阶/五阶时间精确算法，${\mathbf{夸德}}_3（\gamma ，{\rho }^{\infty }）$ 和 ${\mathbf{立方体}}_5（\gamma ，{\rho }^{\infty }）$，具有可控制的数值耗散。与众所周知的隐式Runge-Kutta（RK）算法家族相比，这些新开发的算法${\mathbf{夸德}}_3（\gamma ，{\rho }^{\infty }）$ 和 ${\mathbf{立方体}}_5（\gamma ，{\rho }^{\infty }）$可以（a）通过选择某些算法参数来恢复Radau IIA ₃ / RK3和Radau IIA ₅ / RK5方案（$\gamma ，{\rho }^{\infty }$）; （b）获得许多解决方案精度更高的新算法，这些算法优于RK系列算法，并且（c）具有与隐式RK系列算法相似的计算效率。研究了几个单/多自由度（SDOF和MDOF）问题，以验证提出的发展。此外，值得注意的是，拟议的方法还可以使用高阶（不限于二次和三次）基函数来设计更高级的方案，并且可以与高阶空间离散化方法集成在一起，解决空间问题。时间PDE，例如等几何方法，SEM，间断Galerkin（DGM），p版本FEM等。