An optimal control based method for coupling 1D/2D models applied to river hydraulics
Creators: name: Gejadze, Igor
name: Monnier, Jerome
name: Gejadze, Igor
Publisher: name: XVI International Conference on Computational Methods in Water Resources
Year: 2006

description: 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.