Identification of Transport Coefficient Models in Convection-Diffusion Equations

Abstract

A rigorous method is presented for the systematic identification of the structure and the parameters of transport coefficient models in three-dimensional, transient convection-diffusion systems using high resolution measurement data. The transport is represented by a convection term with known convective velocity and a diffusion term with an unknown, generally state-dependent, transport coefficient. The identification of a transport coefficient model constitutes an ill-posed, highly nonlinear inverse problem. In our previous work [29], we presented a novel incremental identification method, which decomposes this inverse problem into easier to handle inverse subproblems. This way, the incremental identification method not only allows for the identification of the structure and the parameters of the model, but also supports the rigorous decision making on the best suited transport model structure. Due to the decomposition approach, the identified transport model structure and parameters are subject to errors. To cope with the error propagation inherent to the incremental method, the present work suggests a model correction procedure as a supplement to the incremental identification method [29], which results in a transport model of higher precision. The correction refers to both, the model structure and parameters. No a priori knowledge on the unknown transport model structure is necessary. The identification approach is numerically illustrated for a three-dimensional, transient convection-diffusion equation which has its origin in the modeling and simulation of energy transport in a laminar wavy film flow.

On reduced modeling of mass transport in wavy falling films

“…Since the considered reduced model makes some assumptions (2D geometry, domain and velocity from LWE) to reduce the computational time, the results will be of course different from a DNS simulation or real measurements. To compensate for modeling errors and improve the model in future works, we may apply the incremental identification methodology, that has been already introduced in previous papers . The main idea is to replace the molecular diffusion coefficient D in Eq.…”

“…However, the DNS of the film fluid dynamics requires huge computational effort and sophisticated numerical tools which limit their use in practice, e.g., to perform comprehensive model identification . Therefore, there is a need for a reduced modeling approach, which is capable of representing the most important properties of the transport phenomena in falling films at a modest computational cost.…”

Identification of space-dependent permeability in nonlinear diffusion equation from interior measurements using wavelet multiscale method

“…Inverse coefficient problems for various nonlinear parabolic equations have been investigated by several authors. In [17][18][19], the authors discuss the numerical identification of parameters in nonlinear convection-diffusion equations. The parameter estimation of a class of nonlinear reaction diffusion equations is studied in [20].…”

An adaptive homotopy method for permeability estimation of a nonlinear diffusion equation

“…Parameter inversions of various nonlinear parabolic equations have been studied extensively [17][18][19][20][21][22][23][24]. Ye and Sun [17], considered the inverse problem of the parabolic equation with the control parameter.…”

“…Mocenni et al [18] investigated the parameter estimation of a class of nonlinear reaction diffusion equations. Numerical identification of parameters in nonlinear convection-diffusion equations were studied in [19][20][21]. The parameters of the nonlinear heat conduction problem considered in [22] are spatially and temperature dependent.…”