A mathematical framework towards a unified set of discontinuous state-phase time operators for computational dynamics

Of general interest here is the time dimension aspect wherein discretized operators in time may be continuous or discontinuous; and of particular interest and focus here is the design of time operators in the context of discontinuous state-phase for computational dynamics applications. Based on a generalized bi-discontinuous time weighted residual formulation, the design leading to a new unified set of hierarchical energy conserving and energy decaying time discretized operators are developed for the first time that are fundamentally useful for dynamic computations. Unlike time discontinuous Galerkin approaches, the design is based upon a time discontinuous Petrov-Galerkin-like approach employing asymptotic series type approximations involving values of derivatives of state variable at the beginning of time step for approximating the state variables. As a consequence, this enables to design methods that have spectral properties corresponding to the diagonal, first sub-diagonal and second sub-diagonal Padé approximations. Thus, A-stable schemes of order 2q, L-stable schemes of order 2q - 1 and 2q - 2 are obtained. The spectral equivalent algorithms to diagonal Padé approximations are energy conserving algorithms. The spectral equivalent algorithms to first and second sub-diagonal Padé approximations are energy decaying algorithms with the property of asymptotic annihilation of the high-frequency response. Since Padé approximations have the lowest relative error the developed schemes are optimal in terms of order of accuracy in time, dissipation, dispersion and zero-order displacement and velocity overshoot characteristics. Additionally, the time operators that are spectrally equivalent to the diagonal, first sub-diagonal, and second sub-diagonal Padé exponential maps naturally inherit a hierarchical structure that are extremely useful for time adaptive computations.