Skip to main content Accessibility help
Hostname: page-component-59b7f5684b-569ts Total loading time: 0.586 Render date: 2022-09-25T12:53:11.550Z Has data issue: true Feature Flags: { "shouldUseShareProductTool": true, "shouldUseHypothesis": true, "isUnsiloEnabled": true, "useRatesEcommerce": false, "displayNetworkTab": true, "displayNetworkMapGraph": false, "useSa": true } hasContentIssue true

Adaptive modeling for free-surface flows

Published online by Cambridge University Press:  22 July 2006

Simona Perotto*
MOX-Modeling and Scientific Computing, Department of Mathematics “F. Brioschi”, Politecnico of Milano, via Bonardi 9, 20133 Milano, Italy.
Get access


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 different nature. Ideally, moving from a hierarchy of hydrodynamic models, one should solve throughout the whole domain the most complex model (with solution $u_{\rm{fine}}$) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional ${\cal F}$, we aim to approximate, within a prescribed tolerance τ, the value ${\cal F}(u_{\rm{fine}})$ by means of the quantity ${\cal F}(u_{\rm{adapted}})$, $u_{\rm{adapted}}$ 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 $|{\cal F}(u_{\rm{fine}}) -{\cal F}(u_{\rm{adapted}})|$ below the tolerance τ. This goal is achieved via a suitable a posteriori 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.

Research Article
© EDP Sciences, SMAI, 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)


Actis, R.L., Szabo, B.A. and Schwab, C., Hierarchic models for laminated plates and shells. Comput. Methods Appl. Mech. Engrg. 172 (1999) 79107. CrossRef
Agoshkov, V.I., Ambrosi, D., Pennati, V., Quarteroni, A. and Saleri, F., Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280299. CrossRef
Agoshkov, V.I., Quarteroni, A. and Saleri, F., Recent developments in the numerical simulation of shallow water equations I: boundary conditions. Appl. Numer. Math. 15 (1994) 175200. CrossRef
Amara, M., Capatina-Papaghiuc, D. and Trujillo, D., Hydrodynamical modelling and multidimensional approximation of estuarian river flows. Comput. Visual. Sci. 6 (2004) 3946. CrossRef
Bangerth, W. and Rannacher, R., Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoust. 9 (2001) 575591. CrossRef
Bažant, Z.P., Spurious reflection of elastic waves in nonuniform finite element grids. Comput. Methods Appl. Mech. Engrg. 16 (1978) 91100. CrossRef
R. Becker and R. Rannacher, An optimal control approach to a posteriori error estimation in finite element methods, in Acta Numerica 2001, A. Iserles Ed., Cambridge University Press, Cambridge, UK (2001).
J.P. Benque, A. Haugel and P.L. Viollet, Numerical methods in environmental fluid mechanics, in Engineering Applications of Computational Hydraulics, M.B. Abbott and J.A. Cunge Eds., Vol. II (1982).
Braack, M. and Ern, A., A posteriori control of modeling errors and discretizatin errors. Multiscale Model. Simul. 1 (2003) 221238. CrossRef
Ph. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978).
J.M. Cnossen, H. Bijl, B. Koren and E.H. van Brummelen, Model error estimation in global functionals based on adjoint formulation, in International Conference on Adaptive Modeling and Simulation, ADMOS 2003, N.-E. Wiberg and P. Díez Eds., CIMNE, Barcelona (2003).
A. Ern, S. Perotto and A. Veneziani, Finite element simulation with variable space dimension. The general framework (2006) (in preparation).
Feistauer, M. and Schwab, C., Coupling of an interior Navier-Stokes problem with an exterior Oseen problem. J. Math. Fluid. Mech. 3 (2001) 117. CrossRef
L. Formaggia and A. Quarteroni, Mathematical Modelling and Numerical Simulation of the Cardiovascular System, in Handbook of Numerical Analysis, Vol. XII, North-Holland, Amsterdam (2004) 3–127.
Formaggia, L., Nobile, F., Quarteroni, A. and Veneziani, A., Multiscale modelling of the circolatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 7583. CrossRef
M.B. Giles and N.A. Pierce, Adjoint equations in CFD: duality, boundary conditions and solution behaviour, in 13th Computational Fluid Dynamics Conference Proceedings (1997) AIAA paper 97–1850.
Giles, M.B. and Süli, E., Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145236. CrossRef
E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996).
Griewank, A. and Walther, A., Revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM T. Math. Software 26 (2000) 1945. CrossRef
Harari, I., Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Engrg. 140 (1997) 3958. CrossRef
J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Volume I. Springer-Verlag, Berlin (1972).
G.I. Marchuk, Adjoint Equations and Analysis of Complex Systems. Kluwer Academic Publishers, Dordrecht (1995).
G.I. Marchuk, V.I. Agoshkov and V.P. Shutyaev, Adjoint equations and perturbation algorithms in nonlinear problems. CRC Press (1996).
S. Micheletti and S. Perotto (2006) (in preparation).
E. Miglio, S. Perotto and F. Saleri, A multiphysics strategy for free-surface flows, Domain Decomposition Methods in Science and Engineering, R. Kornhuber, R.H.W. Hoppe, J. Périaux, O. Pironneau, O. Widlund, J. Xu Eds., Springer-Verlag, Lect. Notes Comput. Sci. Engrg. 40 (2004) 395–402.
Miglio, E., Perotto, S. and Saleri, F., Model coupling techniques for free-surface flow problems. Part I. Nonlinear Analysis 63 (2005) 18851896. CrossRef
Oden, J.T. and Prudhomme, S., Estimation of modeling error in computational mechanics. J. Comput. Phys. 182 (2002) 469515. CrossRef
Oden, J.T. and Vemaganti, K.S., Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms. J. Comput. Phys. 164 (2000) 2247. CrossRef
Oden, J.T. and Vemaganti, K.S., Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. II. A computational environment for adaptive modeling of heterogeneous elastic solids. Comput. Methods Appl. Mech. Engrg. 190 (2001) 60896124. CrossRef
Oden, J.T., Prudhomme, S., Hammerand, D.C. and Kuczma, M.S., Modeling error and adaptivity in nonlinear continuum mechanics. Comput. Method. Appl. M. 190 (2001) 66636684. CrossRef
A. Quarteroni and L. Stolcis, Heterogeneous domain decomposition for compressible flows, in Proceedings of the ICFD Conference on Numerical Methods for Fluid Dynamics, M. Baines and W.K. Morton Eds., Oxford University Press, Oxford (1995) 113–128.
A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford University Press Inc., New York (1999).
Schulz, M. and Steinebach, G., Two-dimensional modelling of the river Rhine. J. Comput. Appl. Math. 145 (2002) 1120. CrossRef
Stein, E. and Ohnimus, S., Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 363385. CrossRef
G.S. Stelling, On the construction of computational models for shallow water equations. Rijkswaterstaat Communication 35 (1984).
C.B. Vreugdenhil, Numerical Methods for Shallow-Water Flows. Kluwer Academic Press, Dordrecht (1998).
G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York (1974).
F.W. Wubs, Numerical solution of the shallow-water equations. CWI Tract, 49, F.W. Wubs Ed., Amsterdam (1988).

Save article to Kindle

To save this article to your Kindle, first ensure is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the or variations. ‘’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

Adaptive modeling for free-surface flows
Available formats

Save article to Dropbox

To save this article to your Dropbox account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Dropbox account. Find out more about saving content to Dropbox.

Adaptive modeling for free-surface flows
Available formats

Save article to Google Drive

To save this article to your Google Drive account, please select one or more formats and confirm that you agree to abide by our usage policies. If this is the first time you used this feature, you will be asked to authorise Cambridge Core to connect with your Google Drive account. Find out more about saving content to Google Drive.

Adaptive modeling for free-surface flows
Available formats

Reply to: Submit a response

Please enter your response.

Your details

Please enter a valid email address.

Conflicting interests

Do you have any conflicting interests? *