Temporal moments revisited: Why there is no better way for physically based model reduction in time

Authors

  • P. C. Leube,

    Corresponding author
    1. Institute for Modelling Hydraulic and Environmental Systems (LH2), SimTech, University of Stuttgart,Stuttgart,Germany
      Corresponding author: P. C. Leube, Institute for Modelling Hydraulic and Environmental Systems (LH2), SimTech, University of Stuttgart, Pfaffenwaldring 61, DE-70569 Stuttgart, Germany. (philipp.leube@iws.uni-stuttgart.de)
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  • W. Nowak,

    1. Institute for Modelling Hydraulic and Environmental Systems (LH2), SimTech, University of Stuttgart,Stuttgart,Germany
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  • G. Schneider

    1. Institute for Analysis, Dynamics and Modelling, SimTech, University of Stuttgart,Stuttgart,Germany
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Corresponding author: P. C. Leube, Institute for Modelling Hydraulic and Environmental Systems (LH2), SimTech, University of Stuttgart, Pfaffenwaldring 61, DE-70569 Stuttgart, Germany. (philipp.leube@iws.uni-stuttgart.de)

Abstract

[1] Many hydro(geo)logical problems are highly complex in space and time, coupled with scale issues, variability, and uncertainty. Especially time-dependent models often consume enormous computational resources, but model reduction techniques can alleviate this problem. Temporal moments (TM) offer an approach to reduce the time demands of transient hydro(geo)logical simulations. TM reduce transient governing equations to steady state and directly simulate the temporal characteristics of the system, if the equations are linear and coefficients are time independent. This is achieved by an integral transform, projecting the dynamic system response onto monomials in time. In comparison to classical approaches of model reduction that involve orthogonal base functions, however, the monomials for TM are nonorthogonal, which might impair the quality and efficiency of model reduction. Thus, we raise the question of whether there are more suitable temporal base functions than the monomials that lead to TM. In this work, we will derive theoretically that there is only a limited class of temporal base functions that can reduce hydro(geo)logical models. By comparing those to TM we conclude that, in terms of gained efficiency versus maintained accuracy, TM are the best possible choice. While our theoretical results hold for all systems of linear partial or ordinary differential equations (PDEs, ODEs) with any order of space and time derivatives, we illustrate our study with an example of pumping tests in a confined aquifer. For that case, we demonstrate that two (four) TM are sufficient to represent more than 80% (90%) of the dynamic behavior, and that the information content strictly increases with increasing TM order.

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