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1 A new iterative technique for solving nonlinear coupled equations arising from nuclear waste transport processes H. HOTEIT 1,2, Ph. ACKERER 2, R. MOSE 2,3 1 IRISA-INRIA, Rennes 2 Institut de Mécanique des Fluides et des Solides, IMFS, Strasbourg 3 Ecole Nationale du Génie de l'Eau et de l'Environnement, ENGEES, Strasbourg 34 ème Congrès National d'Analyse Numérique 27 Mai - 31 Mai 2002

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2 Outline Mathematical model of the transport processes. Numerical methods: Mixed Hybride Finite Element method (MHFE); Discontinuous Galerkin method (DG). Linearization techniques: Picard (fixed point) method; Newton-Raphson method. Some numerical results.

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3 Transport Processes The transport process concerns an isolated nuclide chain : with the following transport mechanisms : advection, dispersion/diffusion ; mass production/reduction ; precipitation/dissolution ; simplified chemical reactions (sorption).

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4 Mathematical model Transport equation S k is a nonlinear precipitation/dissolution term

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5 Numerical methods Operator splitting technique is used by coupling Diffusion/dispersion by MHFEM Advection by DGM Linearization is done by using Picard (Fixed Point) method Newton-Raphson method

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6 MHFE Advantages mass is conserved locally ; the state head and its gradient are approximated simultaneously ; velocity is determined everywhere due to Raviart-Thomas space functions; full tensors of permeability are easily approximated ; Fourier BC are easily handled ; it can be simply extended to unstructured 2D and 3D grids ; the linear system to solve is positive definite. Disadvantages scheme is non monotone ; number of degrees of freedom=number of sides (faces).

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7 DGM Advantages mass is conserved locally ; satisfies a maximum principle (conserves the positively of the solution) ; can capture shocks without producing spurious oscillation ; ability to handle complicated geometries ; simple treatment of boundary conditions. Disadvantages limited choice of the time-step (explicit time discretization) ; slope (flux) limiting operator stabilize the scheme but creates small amount of numerical diffusion.

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8 Linearization by the Picard method The transport system is rewritten in the form where,

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9 Linearization by the Picard method The (m+1)th step of the Picard-iteration process Stopping criteria

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10 Linearization by the Picard method Convergence needs very small time steps, otherwise : Residual errors for C and F

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11 Coupling Picard and Newton-Raphson methods Define the residual function By using Taylor’s approximation, we get By simple differentiating, we obtain

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12 Coupling Picard and Newton-Raphson methods The iterative process Time steps

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13 Coupling Picard and Newton-Raphson methods Convergence is attained even with bigger time steps (20 times bigger)

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14 Some numerical results Repository siteNetwork of alveolusElementary cell The repository is made up of a big number of alveolus. Computation is made on an elementary cell. Periodic boundary conditions are used.

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15 10 6 years10 5 years10 4 years

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16 Precipitated and dissolved mass in the domain Mass balance in the domain Relative error after 10 6 years

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17 Conclusion Coupling DG and MHEF methods to solve a transport equation with nonlinear precipitation /dissolution function. By using the Picard method, small time steps should be considered otherwise no convergence is attained. Coupling Picard and Newton-Raphson methods Newton-Raphson methods is used for solid phase equation. Picard method methods is used for the transport equation. Convergence is attained even with bigger time steps (20 times bigger).

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