Overview and Novel Insights into Implicit/Explicit Composite Time Integration Type Methods—Fall Under the RK: No Ifs, Ands, or Buts

In this paper, we provide an overview and describe some novel insights, designs and developments, and demonstrate the general methodology and approach towards computationally convenient frameworks for time-dependent problems. The focus is upon time-dependent algorithms and designs, which belong to the Runge–Kutta (RK) type methods and off springs thereof, but termed and appear under the terminology, ‘composite time integration’ in the literature. We particularly revisit and as an illustration, an arbitrary and targeted selection of existing 17 implicit and 10 explicit composite time integration algorithms cited in the literature over the past two decades are synthesized and analyzed. It is studied and next proved that the composite time integration methods have the identical algorithmic structure, namely, the algorithmic properties for the primary algorithmic measures involving stability, accuracy, convergence, and error analysis, as in the original RK method. Consequently, although they may demonstrate some added utility under certain conditions, however small or large, these improvements relate to the secondary measures such as numerical dissipation, dispersion, overshooting, and aliasing behavior for structural dynamics applications. In principle, the math community does not really strictly dwell upon and/or is not concerned about such secondary measures unlike the computational mechanics community which seeks to stress on secondary measures as a metric however limited or not the utility and significance. Traditionally in structural and multi-body dynamics, in comparison to second-order time accurate linear multi-step methods, and/or equivalent single-step methods, the second-order RK type methods are not as popular nor the preferred choice in commercial software. Along similar lines, the composite time integration results in inheriting the same attribute, and the same argument holds for composite methods as in RK methods. Besides describing and highlighting the overall theoretical considerations and designs for composite type integration methods, numerous equivalences are demonstrated of the various existing composite time integration methods, both implicit and explicit compared with those of the RK type DIRK formulations, and the identical Butcher tables are also shown in detail. Simple numerical examples are purposely presented so that researchers can mimic the basic concepts and readily validate the various algorithms and results.