An optimal control based method for coupling 1D/2D ...

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XVI International Conference on Computational Methods in Water Resources (CMWR-XVI) Ingeniørhuset

An optimal control based method for coupling 1D/2D models applied to river hydraulics
Author:Igor Gejadze <> (LMC-IMAG)
Jerome Monnier <> (LMC-IMAG)
Presenter:Igor Gejadze <> (LMC-IMAG)
Date: 2006-06-18     Track: Special Sessions     Session: Data assimilation in water resources modelling

Although computer power has been steadily growing in recent years, operational hydrological models describing river networks are still based on the 1D shallow water equations (SWE) or the Saint-Venant equations. Essentially two-dimensional situations ("storage areas", for example) are represented by source terms, which are computed using empirical expressions. Obviously, this approach suffers accuracy limitations and tells us nothing about 2D flow patterns in the area of interest. This justifies the use of limited area 2D SWE models, coupled in a certain way with the 1D-net global model. These 2D models can be viewed as 'zooms' allowing us to improve the classical 1D storage area. A method commonly used to deal with this problem is actually the domain decomposition method, when we obtain a set of 1D channels and 2D areas with or without some overlap between them. In this case the complete 1D-net global no longer exists. We proceed from a practical condition that the 1D-net global operational model must stay intact. We also recognize that computational savings from not running the 1D model within 2D local areas are marginal. Thus, we suggest a coupling principle as follows: we keep the overall "unity" of the 1D model, but the source terms to it are estimated via the 2D SWE variables as fluxes through the reference surfaces. The 1D model, in its own turn, provides the key part of the characteristic boundary conditions, which are required to specify a well-posed 2D problem. First we build the coupling procedure known as the "waveform relaxation method", when 1D and 2D models are solved consecutively in the entire time domain providing the necessary information to each other. Then we write coupling conditions in a "weak" form considering the corresponding objective functional, which is minimized using the adjoint method. Numerical experiments show that the optimal control approach basically requires more iterations as compared to the waveform relaxation method, but allows more accurate results to be obtained. The situation changes in favour of the optimal control approach when we have to assimilate data into such a combined model. That is, the residuals between model outputs and measured values constitute terms into the generalized objective functional in addition to those terms that originate from the coupling problem. This functional is minimized using the same minimization algorithm. Numerical experiments show that the joint assimilation-coupling process converges with a similar speed as the assimilation process performed for the coupled model, i.e. the computationally expensive inner coupling loop can be avoided! Another advantage of this approach is that the 1D and the 2D models are actually independent of each other and, therefore, one can use available standard software. Of course, all the models can be run in parallel. We shall call this approach the domain decomposition of the data assimilation problem. In numerical tests we consider a toy flooding event that involves overflowing of the main channel and a moving front travelling over previously dry areas. This is a case which cannot be described in the 1D network.