Time discretized operators. Part 1: Towards the theoretical design of a new generation of a generalized family of unconditionally stable implicit and explicit representations of arbitrary order for computational dynamics

The objectives and motivation of the exposition presented in Parts 1 and 2 are to fundamentally describe from new perspectives the generic design, development and formal theory towards a new generation of a generalized family of time discretized operators possessing excellent algorithmic attributes as related to the notion of stability and accuracy, and, which also closely mimic the properties of the exact solution of dynamic systems including providing practically useful forms. Subsequently, avenues that also lead to various other time discretized operators are described. In Part 1, the generalized theoretical developments and representations of time discretized operators which theoretically inherit Nth order accuracy for dynamic systems is first designed and proposed which encompass both implicit and explicit unconditionally stable representations of the time discretized operators. Whereas Part 1 primarily focuses on the fundamental theoretical developments and rigor of the generalized representations as related to stability, accuracy and the like, in Part 2 we specifically focus attention towards practical second-order time accurate representations including also assessing the algorithmic attributes and extensions to nonlinear situations. Also described as particular cases are the consequences leading to various other time discretized operators. Simple illustrative numerical examples are then presented to demonstrate the excellent algorithmic and numerical properties of selected implicit and explicit secondorder time discretized operators which closely mimic the properties of the exact solutions including nonlinear dynamic response situations.