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# A Mathematical Framework Towards a Unified Set of Discontinuous State-Phase Hierarchical Time Operators for Computational Dynamics

Ph. D., Research Associate, Department of Mechanical Engineering, 111 Church Street S.E., University of Minnesota, MN, U.S.A., ramdev@me.umn.edu, Ph: 612-626-8101, Fax: 612-626-1596.
Professor and Technical Director, Mechanical Engineering Department/Army High Performance Computing Research Center, 111 Church Street S.E., Technical Director, AHPCRC, University of Minnesota, MN, U.S.A. ktamma@tc.umn.edu, Ph: 612-626-8102, Fax: 612-626-1596.

Computer Modeling in Engineering & Sciences 2003, 4(1), 103-118. https://doi.org/10.3970/cmes.2003.004.103

### Abstract

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 discretized 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 dissipating time discretized operators are developed for the first time that are fundamentally useful for time adaptive computations for dynamic problems. Unlike time discontinuous Galerkin approaches, the design is based upon a time discontinuous Petrov-Galerkin-like approach employing an asymptotic series type approximations for the state variables involving derivatives at the beginning of the time step. As a consequence, this enables to design methods that have spectral properties corresponding to the diagonal, first sub-diagonal and second sub-diagonal Pad\'{e} entries. Thus, \emph {A}-stable schemes of order$2q$, \emph {L}-stable schemes of order$2q-1$ and$2q-2$ are obtained. The spectral equivalent algorithms to the diagonal Pad\'{e} entry are energy conserving algorithms. The spectral equivalent algorithms to the first and second sub-diagonal Pad\'{e} entries are energy dissipating algorithms with the property of asymptotic annihilation of the high frequency response. Additionally, these time operators naturally inherit a hierarchical structure that are extremely useful for time adaptive computations. Moreover, since Pad\'{e} entries 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.

### Keywords

Time discretized discontinuous operators, Generalized weighted residual approach, Hierarchical Time Formulations, Computational Dynamics

, R., , K. (2003). A Mathematical Framework Towards a Unified Set of Discontinuous State-Phase Hierarchical Time Operators for Computational Dynamics. CMES-Computer Modeling in Engineering & Sciences, 4(1), 103–118.

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