1. Combining the VP and EVP models to solve the sea ice momentum equation
Jean-François Lemieux
Dana Knoll, David Holland, Elizabeth Hunke
AOMIP workshop
20 October 2010

2. Pros and cons of the VP and EVP models
VP solution!
Implicit approach (no stability issue)
Issues with parallelization
Slow numerical convergence VP
EVP
Naturally suited for parallelization
Easy to implement
Undamped elastic waves

3. The goal: combine the VP and EVP models
VP+EVP

4. The momentum equation where
We want to solve this implicitly at time t:

5. The nonlinear system of equations

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

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

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

9. The preconditioning operator
Should be efficient
Given a Krylov vector y, we get the vector z as

10. Jacobi as a preconditioner
Recall we want to solve:
We can use A for the preconditioning step:
do p=1, 10
enddo

11. EVP as a preconditioner
Recall that with Jacobi we have:
With the EVP, we time-step the following:
do p=1, 10
enddo

12. Computational efficiency

13. Is our treatment of the inertial term inconsistent?
With Jacobi:
With EVP:

14. EVP as a preconditioner (2nd try)
Recall that with Jacobi we have:
With the EVP, we now time-step the following:
do p=1, Np
enddo

15. Conclusions Possible to recast EVP as a preconditioner
EVP as a preconditioner is however less efficient than a Jacobi iteration
We think there is an inconsistency in our mathematical derivation
Please call if you know what’s wrong!!!
Should I remove the 1st conclusion.

16. Thank you!
With the tanh. Quadratic in the vicinity of the solution.

17. With the tanh. Quadratic in the vicinity of the solution.

18. Viscous-plastic formulationq
-P/2 .
ridge
lead
Advantage of yield curve: need two stresses to represent the failure. Eps are the strain rates – function of the velocity. Very little tensile strength. Can resist large stresses before yielding in compression. Hence, for a slightly divergent field, the stress is close to zero while it can be very high for a slightly convergent field.
Stresses should be on the yield curve.
Hibler, 1979

19. The model Dynamic/thermodynamic VP rheology, ellipse (Hibler, 1979)
Domain: Arctic, North Atlantic and CAA
Resolutions: 10, 20, 40 or 80 km (C-grid)
Coupled to a slab ocean model
Forcing:
- geostrophic winds NCEP/NCAR (6h)
- climatological currents
Advection of momentum is neglected. Thermo model also but we are here interested in the dynamic.

20. Computational gain of JFNK over the standard solver
Quality of
approx.
solution
Cpu gains are multiplicative

21. Typical shear deformation field (10 km)

22. Failures of the two solvers

23. In matrix form… Newton method Standard method (iteration k)
F(u)=A(u)u-b=0
Standard method
(iteration k)
Newton method
(iteration k)
A(uk-1)uk = b
J(uk-1)duk = -F(uk-1)
uk = uk-1 - A-1(uk-1)F(uk-1)
uk = uk-1 - J-1(uk-1)F(uk-1)
uk = uk-1 - (A(uk-1)+G(uk-1))-1F(uk-1)

24. Failure of the line search method