A new iterative technique for solving nonlinear coupled equations arising

1. A new iterative technique for solving nonlinear coupled equations arising from nuclear waste transport processes H. HOTEIT 1,2, Ph. ACKERER2, R. MOSE2,3 1IRISA-INRIA, Rennes 2Institut de Mécanique des Fluides et des Solides, IMFS, Strasbourg 3Ecole Nationale du Génie de l'Eau et de l'Environnement, ENGEES, Strasbourg 34ème 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 Sk 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 • The repository is made up of a big number of alveolus. • Computation is made on an elementary cell . • Periodic boundary conditions are used . Repository site Network of alveolus Elementary cell

15. 104 years 105 years 106 years

16. Precipitated and dissolved mass in the domain Mass balance in the domain Relative error after 106 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).