Design/analysis of GEGS4-1 time integration framework with improved stability and solution accuracy for first-order transient systems

Abstract In this work, the fundamental design procedure, termed as Algorithms by Design, is exploited to establish novel explicit algorithms under the umbrella of linear multi-step (LMS) methods for first-order linear and/or nonlinear transient systems with second-/third-/fourth-order accuracy features. To this end, we focus on developing and designing General Explicit time integration algorithms in an advanced algorithmic fashion typical of the Generalized Single-Step Single-Solve framework for the first-order transient system (GEGS4-1), in which the original GS4-1 has been acknowledged to encompass a wide variety of implicit LMS algorithms of second-order accuracy developed over the past few decades. In contrast to the existing explicit LMS family of algorithms (specifically, second-/third-/fourth-order Adams-Bashforth methods), the proposed algorithmic framework is a single-step formulation and is proved to significantly improve stability and solution accuracy with rigor via mathematical derivations and numerical demonstrations; Moreover, it does not need any additional numerical techniques, such as Runge-Kutta method, for the starting procedure. New/Optimized algorithms can be generated in the proposed framework to circumvent the stability and accuracy limitation with respect to the classical LMS family (not multi-stage method), which is most useful for practical applications. Most significantly, the proposed method readily provides a promising and controllable trade off between stability and accuracy. Specifically, (i) with different selections of free algorithmic parameters, one can recover second-order Adams-Bashforth and Taylor-Galerkin algorithms with critical stability frequency Ω s = λ Δ t c r = 1 , third-order Adams-Bashforth algorithm with Ω s = 6 11 ≈ 0.5455 , and fourth-order Adams-Bashforth algorithm with Ω s = 0.3 ; (ii) new algorithms are originated from the proposed method with improved stability (such as second-order GEGS4-1 with Ω s = 1.2 and/or 1.5, third-order GEGS4-1 with Ω s =1, 1.2, and/or 1.5, and fourth-order GEGS4-1 with Ω s =0.6, 0.8, and/or 1.0) and solution accuracy are presented. Both single-degree of freedom (SDOF) and multi-degree of freedom (MDOF) problems are utilized to validate and demonstrate the ability of proposed algorithmic framework.