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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
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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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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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Mathematical model
Transport equation
Sk is a nonlinear precipitation/dissolution term
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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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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 freedomnumber of sides
  (faces).

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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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Linearization by the Picard method
The transport system is rewritten in the form
where,
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Linearization by the Picard method
The (m1)th step of the Picard-iteration process
Stopping criteria
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Linearization by the Picard method
Convergence needs very small time steps,
otherwise
Residual errors for C and F
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Coupling Picard and Newton-Raphson methods
Define the residual function
By using Taylors approximation , we get
By simple differentiating, we obtain
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Coupling Picard and Newton-Raphson methods
The iterative process
Time steps
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Coupling Picard and Newton-Raphson methods
Convergence is attained even with bigger time
steps (20 times bigger)
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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
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106 years
105 years
104 years
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Precipitated and dissolved mass in the domain
Mass balance in the domain
Relative error after 106 years
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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).