Design of order-preserving algorithms for transient first-order systems with controllable numerical dissipation

Abstract

SUMMARYUsing a new design procedure termed as Algorithms by Design, which we have successfully introduced in our previous efforts for second-order systems, alternatively, we advance in this exposition, the design and development of a computational framework that permits order-preserving second-order time accurate, unconditionally stable, zero-order overshooting behavior, and features with controllable numerical dissipation and dispersion via a family of algorithms for effectively solving transient first-order systems. The key feature is the incorporation of a spurious root to introduce controllable numerical dissipation while preserving second-order accuracy (order-preserving feature) resulting in a two-root system, namely, the principal root ( 1∞ ) and a spurious root ( 2∞ ). In contrast to the classical Trapezoidal family of algorithms which are the most popular, the present framework has the same order of computational complexity, but a higher payoff that is a significant advance to the field for tackling a wide class of applications dealing with first-order transient systems. We also present the special case with selection of 1∞ = 1 and any 2∞ leading to the design of a family of generalized single-step single-solve [GS4-1] algorithms recovering the Crank-Nicolson method at one end ( 2∞ = 1) and the Midpoint Rule at the other end ( 2∞ = 0) and anything in between, all of which have spectral radius features resembling that of the Crank-Nicolson method. More interestingly, with the particular choice of 1∞ = 2∞ = 0, the developed framework additionally inherits L-stable features. We illustrate the successful design of the developed GS4-1 framework using two simple illustrative numerical examples.

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A novel design of an isochronous integration [iIntegration] framework for first/second order multidisciplinary transient systems

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Isochronous Explicit Time Integration Framework: Illustration to Thermal Stress Problems Involving Both First- and Second-Order Transient Systems

“…Performing the set of assignments described by the adapting process yields the explicit GS4-1 Downloaded by [University of Tennessee, Knoxville] at 15:01 22 December 2014 Table 1 The adapting process for properly using the GS4-2 framework as the GS4-1 framework For nodal vectors n ∈ 0 1 n t Treat a n in the GS4-2 framework as n in the first order system Treat v n in the GS4-2 framework as s n in the first order system Neglect q n in the GS4-2 framework (i.e., dummy variable) For parameters Set s in the GS4-2 framework as s in the GS4-1 framework Set max in the GS4-2 framework to equal to unity Set min in the GS4-2 framework as in the GS4-1 framework For U0 Family Additionally require s = min and yield the GS4-1 framework without the selective control feature For V0 Family Choose s min and yield the GS4-1 framework with/without the selective control feature family of algorithms, which is the explicit version of the family of algorithms presented in [4]. That is, this "adaptation process" is carefully formulated using the knowledge of how the two frameworks are related to each other, such that the explicit GS4-2 framework can be correctly adapted to yield the corresponding explicit GS4-1 framework while preserving the physical interpretation and criteria of the algorithmic parameters in both frameworks.…”

“…The unified implicit time integrator framework, hereby termed the implicit iIntegrator unifies the implicit GS4-1 framework [4] (recently developed for first-order systems) and the implicit GS4-2 framework [5] (previously developed for second-order systems) for simultaneous use in both systems with optimal algorithms; with secondorder accuracy, unconditional stability, with minimal numerical dissipation and dispersion, and no more than zero-order overshoot behavior(s). In contrast to all previous developments, the unified implicit iIntegrator framework contains the implicit GS4-1 framework with its feature of selective control of high-frequency damping for use in a first-order system.…”