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

• Published in: Journal of computational physics 2020-12, Vol.422, p.109763, Article 109763
• Main Authors: Wang, Yazhou, Maxam, Dean, Tamma, Kumar K., Qin, Guoliang
• Format: Article
• Language: English
• Subjects: Accuracy; Algorithms; Computational physics; Control stability; Degrees of freedom; First-order systems; Frequency stability; General explicit algorithms; GS4-1 framework; Improved stability and solution accuracy; LMS algorithms; Nonlinear systems; Runge-Kutta method; Stability analysis; Time integration
• ISSN: 0021-9991; 1090-2716
• DOI: 10.1016/j.jcp.2020.109763

•A general purpose methodology is developed to design and generate low to high order accuracy explicit algorithms.•Focus on generating high-order LMS methods which are suitable for practical applications.•The basic concepts are demonstrated purposely through simple examples validating the purposed theoretical claims.•Proposed GEGS4-1 algorithms have improved stability and solution accuracy in comparison to AB and TG methods.•No special starting procedure is required and high-order explicit algorithms can be readily generated. 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 Ωs=λΔtcr=1, third-order Adams-Bashforth algorithm with Ωs=611≈0.5455, and fourth-order Adams-Bashforth algorithm with Ωs=0.3; (ii) new algorithms are originated from the proposed method with improved stability (such as second-order GEGS4-1 with Ωs=1.2 and/or