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CivilComp Proceedings
ISSN 17593433 CCP: 84
PROCEEDINGS OF THE FIFTH INTERNATIONAL CONFERENCE ON ENGINEERING COMPUTATIONAL TECHNOLOGY Edited by: B.H.V. Topping, G. Montero and R. Montenegro
Paper 102
Numerical Solution of Direct and Inverse Problems in Solute Leaching Modelling M.I. Asensio^{1}, L. Ferragut^{1}, M.S. RodríguezCruz^{2} and M.J. SánchezMartín^{2}
^{1}Department of Applied Mathematics, University of Salamanca, Spain
M.I. Asensio, L. Ferragut, M.S. RodríguezCruz, M.J. SánchezMartín, "Numerical Solution of Direct and Inverse Problems in Solute Leaching Modelling", in B.H.V. Topping, G. Montero, R. Montenegro, (Editors), "Proceedings of the Fifth International Conference on Engineering Computational Technology", CivilComp Press, Stirlingshire, UK, Paper 102, 2006. doi:10.4203/ccp.84.102
Keywords: direct and inverse problems, stabilized finite element methods, LevenbergMarquardt algorithm, leaching of pesticides modelling.
Summary
We present two different onedimensional models describing leaching of solutes in columns of soils: a classical
linear equilibrium model for transport of non volatile solutes [4];
and a nonequilibrium sorption model to represent nonequilibrium
processes during transport, which includes the chemical
nonequilibrium [3], or two sites models, and the physical
nonequilibrium, or two regions models [5]. The state
equations given by these models are basically second order linear parabolic
equations, with advective, convective and reactive terms.
The inverse problem of the parameter adjustment is a challenging task. Any numerical method we use to solve the corresponding optimal control problem implies the computation of the gradient of the cost functional. Instead of a numerical approximation of this gradient, we compute it directly by solving the corresponding adjoint problem which is again a similar second order linear parabolic equation, but backward in time. The cost functional is illconditioned, so we propose a LevenbergMarquardt algorithm [6], which combine the gradient descent and Newton methods. The state and the adjoint state equations are both unsteady linear advectiondiffusionreaction onedimensional equations. The main difficulty in the numerical approximation of these partial differential equations, is the accurate modelling of the interaction between advection, diffusion and reaction processes. This forces the use of small time and space steps, but if we want to fit the parameters with a low computational cost we need to work with coarse meshes, so the solution is to use stabilization techniques. Moreover, if we do not know a priori the relative weight of the diffusive, convective and reactive terms, we need to guarantee a scheme stable for any regimen to assure a right parameter adjustment. We propose a numerical scheme based on the link cutting bubble strategy (LCB) introduced by Brezzi et al. [2] for the steady problem, adapted for the non steady case in [1] where this scheme is compared to more classical stabilization techniques. The key point of this scheme is precisely that it has been designed to work in all regimes, that is, when the diffusion term is dominant, when the advection term is dominant and when the reaction term is dominant. The numerical experiments presented here are based on real data provided by authors in [7]. References
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