Time discretized operators. Part 2: 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 Original Research Article

The new generation of a generalized family of time discretized operators encompassing implicit and explicit representations that are unconditionally stable and which theoretically inherit Nth-order time accurate features developed in Part 1 are restricted here in Part 2 of the exposition to second-order time accurate operators. As such, unconditionally stable implicit representations are first described followed by unconditionally stable explicit representations. The theoretical design leading to computational algorithms with excellent algorithmic attributes for applicability to practical situations are also addressed for both the implicit and explicit unconditionally stable representations of time discretized operators. Attention is first focused on linear problems and extensions to nonlinear situations are subsequently briefly addressed.