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

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.

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).

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

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

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).

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.

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

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

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

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

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

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

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.

years10 5 years10 4 years

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

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).