This work represents a first step towards the simulation of the
motion of water in a complex hydrodynamic configuration, such as
a channel network or a river delta, by means of a suitable
“combination” of different mathematical
models. In this framework a wide spectrum of space and time scales is involved
due to the presence of physical phenomena of
Ideally, moving from a hierarchy of hydrodynamic models, one should solve
throughout the whole domain the most complex model (with solution
) to accurately
describe all the physical features of the problem at hand. In our approach instead,
for a user-defined output functional
, we aim to approximate,
within a prescribed tolerance τ, the value
by means of the quantity
being the so-called adapted solution solving the
simpler models on most of the computational domain while confining the complex ones only on a
restricted region. Moving from
the simplified setting where only two hydrodynamic models, fine and coarse, are considered, we
provide an efficient tool able to automatically select
the regions of the domain where the coarse
model rather than the fine one are to be solved, while guaranteeing
below the tolerance τ.
This goal is achieved via a suitable
modeling error analysis
developed in the framework of a goal-oriented theory.
We extend the dual-based approach provided in
[Braack and Ern,
Multiscale Model Sim.
1 (2003) 221–238],
for steady equations to the case of a generic time-dependent problem.
Then this analysis is specialized to the case we are
interested in, i.e. the free-surface flows simulation, by emphasizing
the crucial issue of the time discretization for both the primal and the dual
problems. Finally, in
the last part of the paper a widespread numerical validation is carried out.