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The G-Scheme: A framework for multi-scale adaptive model reduction
No full textThe numerical solution of mathematical models for reaction systems in general, and reacting flows in particular, is a challenging task because of the simultaneous contribution of a wide range of time scales to the system dynamics. However, the dynamics can develop very-slow and very-fast time scales separated by a range of active time scales. An opportunity to reduce the complexity of the problem arises when the fast/active and slow/active time scales gaps becomes large. We propose a numerical technique consisting of an algorithmic framework, named the G-Scheme, to achieve multi-scale adaptive model reduction along-with the integration of the differential equations (DEs). The method is directly applicable to initial-value ODEs and (by using the method of lines) PDEs. We assume that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. The G-Scheme introduces locally a curvilinear frame of reference, defined by a set of orthonormal basis vectors with corresponding coordinates, attached to this decomposition. The evolution of the curvilinear coordinates associated with the active subspace is described by nonstiff DEs, whereas that associated with the slow and fast subspaces is accounted for by applying algebraic corrections derived from asymptotics of the original problem. Adjusting the active DEs dynamically during the time integration is the most significant feature of the G-Scheme, since the numerical integration is accomplished by solving a number of DEs typically much smaller than the dimension of the original problem, with corresponding saving in computational work. To demonstrate the effectiveness of the G-Scheme, we present results from illustrative as well as from relevant problems
Entropy production at all scales
Full text linkSpatially homogeneous systems are characterized by the simultaneous presence of a wide range of time scales. When the dynamics of such reactive systems develop very-slow and very-fast time scales separated by a range of active time scales, with large gaps in the fast/active and slow/active time scales, then it is possible to achieve multi-scale adaptive model reduction along-with the integration of the governing ordinary differential equations using the G-Scheme framework. The G-Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We derive the expressions that express the direct link between time scales and entropy production by resorting to the estimates provided by the G-Scheme. With reference to a constant volume, adiabatic batch reactor, we compute the contribution to entropy production by the four subspaces. The numerical experiments show that, as indicated by the theoretical derivation, the contribution to entropy production of the fast subspace is of the same magnitude of the error threshold chosen for the numerical integration, and that the contribution of the slow subspace is generally much smaller than that of the active subspace
Entropy production and the g-scheme
Full text linkSpatially homogeneous batch reactor systems are characterized by the simultaneous presence of a wide range of time scales. When the dynamics of such reactive systems develop very-slow and very-fast time scales separated by a range of active time scales, with large gaps in the fast/active and slow/active time scales, then it is possible to achieve multi-scale adaptive model reduction along-with the integration of the governing ordinary differential equations using the G-Scheme framework. The G- Scheme assumes that the dynamics is decomposed into active, slow, fast, and when applicable, invariant subspaces. We computed the contribution to entropy production by the four subspaces, with reference to a constant volume, adiabatic reactor. The numerical experiments indicate that the contributions of the fast and slow subspaces are much smaller than that of the active subspace
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
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