1. Development of an IMEX integration method for sea ice dynamics
J-F Lemieux and Dana Knoll
Woods Hole, MA
October 24, 2012

2. The momentum and continuity equations in 1-D
where
continuity
2 thickness cat. Neglect the thermo terms.
We want to solve these at time levels n= 1, 2, 3,... : Δt
2Δt
3Δt
...

3. The splitting in time approach
Problem du splitting in time. Large change of P can induce a reversal of u.

4. Motivation Lipscomb et al., 2007
Lipscomb: Problem occurs near a coast where strain rates are largest. 9 km model with 1 hour time step.
Lipscomb mitigated this problem by changing the physics. We instead play with the numerics.
Our model also exhibits difficulties: it sometimes has problem converging.
Also in Hutchings et al. , 2004 Strength implicit

5. Fully implicit?
Fully implicit: complicated and a lot of changes to the code.
We explore instead an IMEX approach.
Problem du splitting in time. Large change of P can induce a reversal of u.

6. The JFNK solver
F depends on h and A.
(the residual)

7. F(u) u The Newton method u3 u2 u1 u0
Graphically. Quadratic convergence u u3 u2 u1 u0

8. JFNK JFNK-IMEX do k=1,... do k=1,... solve solve enddo enddo
The two JFNK solvers
JFNK
JFNK-IMEX
do k=1,...
solve
enddo
do k=1,...
solve
enddo

9. The experiment 10 m/s h0=0.5 m A0=0.95 u0=0 m/s 2000 km Δx = 10 km
Simulation of 11 days
Calculate statistics for the last 10 days
Reference solution with a time step of 10 s

10. The reference solution after 11 days

11. Newton iterations and CPU time as a function of Δt

12. h minus href after 11 days (Δt =180 min, Dx=10km)

13. RMSE (thickness) after 11 days (10km)

14. Conclusions
IMEX method seems to mitigate the splitting in time instability
IMEX method involves minor modifications to the code
It is more robust, more accurate and more computationally efficient than the splitting in time approach
It remains to be seen how it behaves in more realistic experiments

15. Thank you!!!

16. Comparison of the JFNK and Picard solvers
40 km.
Picard
JFNK
Lemieux et al. 2010

17. The solution is approximated in the subspace:
The JFNK solver
do k=1,...
solve
enddo
The solution is approximated in the subspace:
where
We can approximate J times a vector by:
where ε is a small number

18. We want to solve Picard JFNK do k=1, kmax Solve if stop enddo

19. The Picard (or standard) solver
do k=1,...
solve
enddo

20. The preconditioned FGMRES method
where
The solution is approximated in the subspace:
where

21. Nb of Newton iterations at each time level (30min, 10km)
Dana Imex0 means splitting in time...