The time dimension and a unified mathematical framework for first-order parabolic systems

For the analysis of problems encompassing linear first-order parabolic systems involving the time dimension, the present exposition describes the evolution of and synthesis leading to a general unified mathematical framework and design of computational algorithms. In our previous efforts, various issues and the general classification and characterization of time-discretized operators were addressed, and the theoretical developments emanated from a generalized time-weighted residual philosophy which described the underlying consequences. Toward this end, in this article, for the first time, we provide alternative new perspectives and formalism via the notions of (1) the resulting size of the equation system and (2) the associated number of system solve(s). Although the time-weighted residual philosophy described an approach and the underlying consequences, from the new perspectives, the general design of computational algorithms is outlined in this article. A generalized stability and accuracy limitation theorem is also highlighted for linear transient algorithms encompassing first-order parabolic systems. Characterization as related to computational algorithms pertains to that which not only permits the general classification to be established but also provides the underlying basis for their subsequent design.