ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 40 (2006) no. 3 , p. 469-499
doi : 10.1051/m2an:2006020
URL stable : http://www.numdam.org/item?id=M2AN_2006__40_3_469_0

Classification:  65J15,  65M15,  65M60
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 $ℱ$, we aim to approximate, within a prescribed tolerance $\tau$, the value $ℱ\left({u}_{\mathrm{fine}}\right)$ by means of the quantity $ℱ\left({u}_{\mathrm{adapted}}\right)$, ${u}_{\mathrm{adapted}}$ 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 $|ℱ\left({u}_{\mathrm{fine}}\right)-ℱ\left({u}_{\mathrm{adapted}}\right)|$ 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.

### Bibliographie

[1] R.L. Actis, B.A. Szabo and C. Schwab, Hierarchic models for laminated plates and shells. Comput. Methods Appl. Mech. Engrg. 172 (1999) 79-107. Zbl 0959.74061

[2] V.I. Agoshkov, D. Ambrosi, V. Pennati, A. Quarteroni and F. Saleri, Mathematical and numerical modelling of shallow water flow. Comput. Mech. 11 (1993) 280-299. Zbl 0771.76032

[3] V.I. Agoshkov, A. Quarteroni and F. Saleri, Recent developments in the numerical simulation of shallow water equations I: boundary conditions. Appl. Numer. Math. 15 (1994) 175-200. Zbl 0833.76008

[4] M. Amara, D. Capatina-Papaghiuc and D. Trujillo, Hydrodynamical modelling and multidimensional approximation of estuarian river flows. Comput. Visual. Sci. 6 (2004) 39-46. Zbl pre02132406

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

[6] Z.P. Bažant, Spurious reflection of elastic waves in nonuniform finite element grids. Comput. Methods Appl. Mech. Engrg. 16 (1978) 91-100. Zbl 0384.73016

[7] 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). MR 2009692 | Zbl 1105.65349

[8] 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).

[9] M. Braack and A. Ern, A posteriori control of modeling errors and discretizatin errors. Multiscale Model. Simul. 1 (2003) 221-238. Zbl 1050.65100

[10] Ph. Ciarlet, The Finite Element Method for Elliptic Problems. North-Holland Publishing Company, Amsterdam (1978). MR 520174 | Zbl 0383.65058

[11] 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).

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

[13] M. Feistauer and C. Schwab, Coupling of an interior Navier-Stokes problem with an exterior Oseen problem. J. Math. Fluid. Mech. 3 (2001) 1-17. Zbl 0991.35061

[14] 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.

[15] L. Formaggia, F. Nobile, A. Quarteroni and A. Veneziani, Multiscale modelling of the circolatory system: a preliminary analysis. Comput. Visual. Sci. 2 (1999) 75-83. Zbl 1067.76624

[16] 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.

[17] M.B. Giles and E. Süli, Adjoint methods for PDEs: a posteriori error analysis and postprocessing by duality. Acta Numerica 11 (2002) 145-236. Zbl 1105.65350

[18] E. Godlewski and P.A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). MR 1410987 | Zbl 0860.65075

[19] A. Griewank and A. Walther, Revolve: an implementation of checkpointing for the reverse or adjoint mode of computational differentiation. ACM T. Math. Software 26 (2000) 19-45. Zbl 1137.65330

[20] I. Harari, Reducing spurious dispersion, anisotropy and reflection in finite element analysis of time-harmonic acoustics. Comput. Methods Appl. Mech. Engrg. 140 (1997) 39-58. Zbl 0898.76058

[21] J.L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Volume I. Springer-Verlag, Berlin (1972). MR 350177 | Zbl 0223.35039

[22] G.I. Marchuk, Adjoint Equations and Analysis of Complex Systems. Kluwer Academic Publishers, Dordrecht (1995). MR 1336381 | Zbl 0817.34002

[23] G.I. Marchuk, V.I. Agoshkov and V.P. Shutyaev, Adjoint equations and perturbation algorithms in nonlinear problems. CRC Press (1996). MR 1413046

[24] S. Micheletti and S. Perotto (2006) (in preparation).

[25] 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. Zbl pre02143570

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

[27] J.T. Oden and S. Prudhomme, Estimation of modeling error in computational mechanics. J. Comput. Phys. 182 (2002) 469-515. Zbl 1053.74049

[28] J.T. Oden and K.S. Vemaganti, 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 0992.74072

[29] J.T. Oden and K.S. Vemaganti, 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 1030.74051

[30] J.T. Oden, S. Prudhomme, D.C. Hammerand and M.S. Kuczma, Modeling error and adaptivity in nonlinear continuum mechanics. Comput. Method. Appl. M. 190 (2001) 6663-6684. Zbl 1012.74081

[31] 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. Zbl 0869.76067

[32] A. Quarteroni and A. Valli, Domain decomposition methods for partial differential equations. Oxford University Press Inc., New York (1999). MR 1857663 | Zbl 0931.65118

[33] M. Schulz and G. Steinebach, Two-dimensional modelling of the river Rhine. J. Comput. Appl. Math. 145 (2002) 11-20. Zbl 1072.86001

[34] E. Stein and S. Ohnimus, Anisotropic discretization- and model-error estimation in solid mechanics by local Neumann problems. Comput. Methods Appl. Mech. Engrg. 176 (1999) 363-385. Zbl 0954.74072

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

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

[37] G.B. Whitham, Linear and Nonlinear Waves. Wiley, New York (1974). MR 483954 | Zbl 0373.76001

[38] F.W. Wubs, Numerical solution of the shallow-water equations. CWI Tract, 49, F.W. Wubs Ed., Amsterdam (1988). MR 957529 | Zbl 0653.76011