Single Field Form
Consider the semi-discretized forms of equation systems obtained for dynamic field problems (say in a finite element sense) following the usual space discretization procedures:
For a given 
since in general, we have a residual error, we wish to minimize this error.
As such, assuming an arbitrary virtual field or weighted time field, 
, for enacting the time discretization process, the above semi-discretized equation system can be cast into the form:
To provide a fundamental insight into the underlying theoretical basis, and to help shed light on providing avenues for the development and characterization, and thereby leading to a wide variety of time discretized operators, following Tamma et al. [88, 81, 8, 106], we propose to first reduce in a mathematically consistent manner the above equation system (1.17) so as to first yield the adjoint operator of the original semi-discretized dynamic field system. Accordingly,
where we define 
as the adjoint operator of the given original second-order single-field form of the semi-discretized dynamic equation system.
Two Field Form
Unlike the aforementioned efforts employing a single-field form, we also focus attention on transforming the second-order differential equation (1.16) into two first-order differential equations for convenience in providing alternate theoretical design developments and characterization.
Letting 
and 
in equation (1.16), the equivalent first-order representation is given as:
where
The development of time discretized operators which emanate under the umbrella and framework and explained via a time weighted philosophy associated with the two-field form of representation are also described.
As such, for illustration, a Galerkin time weighted residual representation in the time domain 
is employed for the two-field form as,
where 
are the arbitrary weighted time fields.
Emanating from the two-field from and using the Green's formula for the equation (1.21) in a consistent manner, yields
where we now define 
as the adjoint operator of the given original first-order two-field form of the semi-discretized dynamic system.
With the exception that 
and consequently 
is undefined, thus far, there are no approximations introduced in developing the reduced forms containing the adjoint operator of the original transient/dynamic semi-discretized system.
We define as that associated with the adjoint operator, wherein f/s denotes the `first' or `second' order system under consideration.
At this juncture, several key observations can be immediately inferred as related to the burden placed upon the virtual or weighted time field, , originally introduced for the time-discretization process and now related to and contained in the reduced form of the original semi-discretized transient/dynamic field problems, respectively.
Remarks
full matrix order, (ii) an diagonal matrix character, or (iii) in a theoretical sense can be degenerated mathematically to approximate instead as a vector or a single-valued scalar time function (other possible are not discussed here).
, the resulting solution can be interpreted as a formal description for as the associated integrating factor of the transient or dynamic system, respectively.
, where 
, and 
the resulting solution can be interpreted as a formal description for as the associated Green's function of the differential equation of the original semi-discretized first or second-order system, respectively. The choice of 
yields on analogous interpretation as in (3) above. , can be either obtained directly or after introducing the notion of employing modal decomposition. The resulting consequences are subsequently addressed. Such a selection via (3) and (4), leads to `explicit representations'. Alternatively, the resulting formulations lead to mostly `implicit representations' from which explicit forms can be constructed. , obtained via (3) or (4) yields an `exact integral operator' (this is subsequently demonstrated later). For this specific selection of , the variations for 
and 
between time levels n and n+1 does not matter.
Any approximations introduced in the `exact integral operator' to approximate the loading functions 
will however, result in that termed as an `approximate integral operator'.
Furthermore, if the virtual or weighted time fields are not selected via (3) or (4), then the resulting approximations can be systematically derived and explained from these consistently. These associated approximations all emanate from a generalized exact virtual or weighted function field, , which can be interpreted as a combined representation of obtained from (3) and (4) resulting in a vector 
or a scalar 
(in a mean sense), from an asymptotic series type approximation. Thereby, now necessitating a corresponding consistent order approximation to be made for the transient or dynamic field variables or respectively, thus leading to those termed as `integration operators'.
This philosophy now readily permits avenues for classification and characterization of time discretized operators via discrete numerically assigned [DNA] algorithmic markers which essentially comprise of: (i) the weighted time fields introduced for enacting the time discretization process, and (ii) the conditions they in turn impose (if any) upon the dependent field variable approximations and the updates in the theoretical design developments. and the resulting consequences in the choices of and evolution of and the conditions they impose upon the approximations for the state variable(s) thereby describe (based on the assumptions invoked), the development of `exact integral operators', `approximate integral operators', or a wide variety of `integration operators' for transient or dynamic field problems, respectively.
Note that analogous developments follow for other time weighted residual satisfaction approaches such as least squares, time discontinuous philosophy and the like (not shown here).
In contrast to past original methods of algorithmic developments available in the literature, a basis for the classification, theoretical design and characterization of transient/dynamic algorithms including providing new avenues which have not been exploited to-date and the recovery of a wide variety of existing time discretized operators is described next. The primary motivation and objectives of the remainder of the present exposition follows next and emanate under the umbrella and framework of a generalized time weighted philosophy. Currently, there exists: (i) the original methods of development of computational algorithms which continue to be employed in various engineering analysis codes, and (ii) various other previous efforts including the weighted residual approach and the like describing alternate formulations which indeed include (to an extent) several of the original developments and also lead to other new formulations and have no doubt provided certain useful generalizations. Nevertheless, these previous efforts existing in the literature which employed the weighted residual type approach and the like in an attempt towards developing certain useful generalizations, fail to strictly enact a mathematically consistent framework of formulation and methodology, thereby, limiting and/or restricting not only the overall scope of the developments and generalizations but also the fundamental understanding of the underlying theoretical basis as well. To an extent these developments indeed provide spectral similarity of algorithms and only to a limited extent they can provide matrix identity of existing algorithms in the process of recovery. In an analogous manner, similar restrictions apply for the other relevant developments available in the literature, although, to a greater extent some of the restrictions have been circumvented. More importantly, none of these past approaches provides a fundamental theoretical basis and a clear understanding and explanation of the underlying weighted time fields themselves which are employed in the development of time discretized operators other than simply alluding to the general idea of introducing parameters which relate to the weighted averages of the time weighted fields. And lastly, these deficiencies strictly limit and hamper the development of other new avenues leading to useful time discretized operators which have not been explored, and prohibit providing a fundamental understanding and relationships amongst the various time discretized operators including a clear explanation of their associations. Furthermore, the basis for approximating the dependent field variables (and whether there exists indeed any need for such approximations) and the updates and the underlying reasons have also not been understood nor clearly explained to-date.
In lieu of the aforementioned considerations, unlike all past efforts available in the literature including our own earlier preliminary efforts towards providing new perspectives, Tamma et al. [106] recently discussed the underlying theoretical basis and framework in describing the evolution of a limited class of time discretized operators, although the developments emanate from a virtual field or time weighted philosophy introduced for enacting the time discretization process. However, our own preliminary efforts were limited in scope and generalizations including characterization of algorithms were restricted to a limited class of spectrally similar algorithms (and recovery of a limited scope of existing algorithms) due to our own lack of providing a more general theoretical basis and framework and limited experiences via the new perspectives we had been developing. Nonetheless, these efforts have served as a useful starting point and now offer a broader vision, new avenues and improved levels of sophistication, thereby, leading to providing a basis for the classification and theoretical design of time discretized operators including their characterization (see [106, 103, 105, 100, 82].
In the present exposition, we first fundamentally address what constitutes classification and characterization of transient/dynamic algorithms, and describe a theory of development/evolution, classification, and theoretical design including characterization of a wide variety of time discretized operators. The underlying answer to characterization being the [DNA] markers which comprise of the burden placed upon the weighted time fields which in turn also additionally dictate the necessary conditions for determining if and what type of approximations for the state vectors, namely, the dependent field variables are needed. As such, a standardized formal theory for the development and evolution of time discretized integral operators and the direct consequences which clearly explain and systematically lead to the evolution of a wide variety of the so-called single-step direct time integration operators from approximations introduced to time integral operators, and then leading to and explaining multi-step integration operators, and subsequently permitting an equivalence to be drawn with a class of finite element in time integration operators and the relevant associations and relationships are particularly explored.
The views and opinions expressed in this page are strictly those of the page author.
The contents of this page have not been reviewed or approved by the University of Minnesota.
