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 ${u}_{\mathrm{fine}}$) to accurately describe all the physical features of the problem at hand. In our approach instead, for a user-defined output functional $\mathcal{F}$, we aim to approximate, within a prescribed tolerance $\tau $, the value $\mathcal{F}\left({u}_{\mathrm{fine}}\right)$ by means of the quantity $\mathcal{F}\left({u}_{\mathrm{adapted}}\right)$, ${u}_{\mathrm{adapted}}$ 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 $|\mathcal{F}\left({u}_{\mathrm{fine}}\right)-\mathcal{F}\left({u}_{\mathrm{adapted}}\right)|$ below the tolerance $\tau $. 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.

Keywords: modeling adaptivity, a posteriori error estimate, goal-oriented analysis, free-surface flows, dual problem, finite elements

@article{M2AN_2006__40_3_469_0, author = {Perotto, Simona}, title = {Adaptive modeling for free-surface flows}, journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique}, pages = {469--499}, publisher = {EDP-Sciences}, volume = {40}, number = {3}, year = {2006}, doi = {10.1051/m2an:2006020}, mrnumber = {2245318}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/m2an:2006020/} }

TY - JOUR AU - Perotto, Simona TI - Adaptive modeling for free-surface flows JO - ESAIM: Modélisation mathématique et analyse numérique PY - 2006 SP - 469 EP - 499 VL - 40 IS - 3 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/m2an:2006020/ DO - 10.1051/m2an:2006020 LA - en ID - M2AN_2006__40_3_469_0 ER -

Perotto, Simona. Adaptive modeling for free-surface flows. ESAIM: Modélisation mathématique et analyse numérique, Volume 40 (2006) no. 3, pp. 469-499. doi : 10.1051/m2an:2006020. http://archive.numdam.org/articles/10.1051/m2an:2006020/

[1] Hierarchic models for laminated plates and shells. Comput. Methods Appl. Mech. Engrg. 172 (1999) 79-107. | Zbl

, and ,[2] Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280-299. | Zbl

, , , and ,[3] Recent developments in the numerical simulation of shallow water equations I: boundary conditions. Appl. Numer. Math. 15 (1994) 175-200. | Zbl

, and ,[4] Hydrodynamical modelling and multidimensional approximation of estuarian river flows. Comput. Visual. Sci. 6 (2004) 39-46.

, and ,[5] Adaptive finite element techniques for the acoustic wave equation. J. Comput. Acoust. 9 (2001) 575-591.

and ,[6] Spurious reflection of elastic waves in nonuniform finite element grids. Comput. Methods Appl. Mech. Engrg. 16 (1978) 91-100. | Zbl

,[7] 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). | MR | Zbl

and ,[8] Numerical methods in environmental fluid mechanics, in Engineering Applications of Computational Hydraulics, M.B. Abbott and J.A. Cunge Eds., Vol. II (1982).

, and ,[9] A posteriori control of modeling errors and discretizatin errors. Multiscale Model. Simul. 1 (2003) 221-238. | Zbl

and ,[10] The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). | MR | Zbl

,[11] 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).

, , and ,[12] Finite element simulation with variable space dimension. The general framework (2006) (in preparation).

, and ,[13] Coupling of an interior Navier-Stokes problem with an exterior Oseen problem. J. Math. Fluid. Mech. 3 (2001) 1-17. | Zbl

and ,[14] Mathematical Modelling and Numerical Simulation of the Cardiovascular System, in Handbook of Numerical Analysis, Vol. XII, North-Holland, Amsterdam (2004) 3-127.

and ,[15] Multiscale modelling of the circolatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75-83. | Zbl

, , and ,[16] Adjoint equations in CFD: duality, boundary conditions and solution behaviour, in 13th Computational Fluid Dynamics Conference Proceedings (1997) AIAA paper 97-1850.

and ,[17] Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145-236. | Zbl

and ,[18] Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR | Zbl

and ,[19] Revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM T. Math. Software 26 (2000) 19-45. | Zbl

and ,[20] Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Engrg. 140 (1997) 39-58. | Zbl

,[21] Non-Homogeneous Boundary Value Problems and Applications. Volume I. Springer-Verlag, Berlin (1972). | MR | Zbl

and ,[22] Adjoint Equations and Analysis of Complex Systems. Kluwer Academic Publishers, Dordrecht (1995). | MR | Zbl

,[23] Adjoint equations and perturbation algorithms in nonlinear problems. CRC Press (1996). | MR

, and ,[24]

and (2006) (in preparation).[25] 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.

, and ,[26] Model coupling techniques for free-surface flow problems. Part I. Nonlinear Analysis 63 (2005) 1885-1896.

, and ,[27] Estimation of modeling error in computational mechanics. J. Comput. Phys. 182 (2002) 469-515. | Zbl

and ,[28] Estimation of local modeling error and goal-oriented adaptive modeling of heterogeneous materials. I. Error estimates and adaptive algorithms. J. Comput. Phys. 164 (2000) 22-47. | Zbl

and ,[29] 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) 6089-6124. | Zbl

and ,[30] Modeling error and adaptivity in nonlinear continuum mechanics. Comput. Method. Appl. M. 190 (2001) 6663-6684. | Zbl

, , and ,[31] 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. | Zbl

and ,[32] Domain decomposition methods for partial differential equations. Oxford University Press Inc., New York (1999). | MR | Zbl

and ,[33] Two-dimensional modelling of the river Rhine. J. Comput. Appl. Math. 145 (2002) 11-20. | Zbl

and ,[34] Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 363-385. | Zbl

and ,[35] On the construction of computational models for shallow water equations. Rijkswaterstaat Communication 35 (1984).

,[36] Numerical Methods for Shallow-Water Flows. Kluwer Academic Press, Dordrecht (1998).

,[37] Linear and Nonlinear Waves. Wiley, New York (1974). | MR | Zbl

,[38] Numerical solution of the shallow-water equations. CWI Tract, 49, F.W. Wubs Ed., Amsterdam (1988). | MR | Zbl

,*Cited by Sources: *