Transient algorithms for heat transfer: General developments and approaches for theoretically generating Nth-order time-accurate operators including practically useful second-order forms

New theoretical ideas and developments describing the fundamental underlying basis for formulating a general family of time discretization operators for first-order parabolic systems emanating from the framework of a generalized time weighted philosophy are first presented which can be broadly classified as pertaining to Type 1, Type 2 and Type 3 family of time discretization operators. As a consequence, the evolution including providing the underlying distinction and the bridging of the relationships between time operators termed as integral operators to the so-called integration operators in time are theoretically developed and demonstrated. The present developments seem to not only provide avenues leading to new algorithms for transient analysis but also provide generalizations and framework to recover a wide variety of existing algorithms. Consequently, under the umbrella of the present framework, a variety of plausible new approaches for generating Nth-order accurate time discretization operators from approximations introduced to Type 1 integral operators income are first described followed by the developments systematically leading to Type 2 time discretization operators, and subsequently to a wide class of Type 3 time integration operators including the recovery of a variety of known existing time integration operators which can be uniquely characterized by Discrete Numerically Assigned (DNA) algorithmic markers. Of the various developments, of noteworthy mention and emphasis here are a new family of L-stable Nth-order Integration Operators (LNInO) of Type 2 for transient computations. Subsequently, some practically useful second-order forms are specifically illustrated and highlighted. The stability and accuracy characteristics are also described for a variety of generated algorithms applicable for transient heat transfer computations. Although the primary focus is on the theoretical developments encompassing linear operators, some simple numerical examples are finally demonstrated to merely illustrate the salient features of the proposed developments.