1. Improving the nonlinear numerical convergence of VP models with the
Jacobian free Newton Krylov method
Jean-François Lemieux
Bruno Tremblay
David Huard
McGill University
Not here to tell you what size of error is acceptable.

2. Outline The nonlinear convergence of existing VP models
Reasons for the slow convergence
The JFNK method
Preliminary results with the JFNK method

3. Outline The nonlinear convergence of existing VP models
Reasons for the slow convergence
The JFNK method
Preliminary results with the JFNK method

4. The model VP rheology, ellipse, normal flow rule (Hibler, 1979)
Domain: Arctic, North Atlantic and CAA
Resolution: 10, 20, 40 or 80 km (C-grid)
Forcing: NCEP 6 hourly varying geostrophic winds and climatological currents.
Acceleration term is included
Advection of momentum is neglected. Thermo model also but we are here interested in the dynamic.

5. The numerical scheme of existing VP models
We want to solve the nonlinear system of eqs:
F(u)=A(u)u-b=0
Provide an initial guess u0
do k=1, kite
Calculate ul = (uk-1+uk-2) / 2
Calculate z(ul), h(ul) and Cd(ul)
Solve A(ul)uk=b with a linear solver
enddo
1 pseudo time step = 2 OL iterations. Results are independent of the linear solver used. Say that it is a fully implicit treatment the way it is written here. Tests were done with a SOR and LSOR solver to verify this.

6. The nonlinear numerical convergence Jan 6 1997 00Z
Dt = 6h
Dx = 10km
FC solution in OL. Limited by machine precision

7. The fully converged (FC) solution Jan 6 1997 00Z
The FC converged solution is certainly an overkilled approximate solution for all the applications.
Dt = 6h
Dx = 10km

8. The error after 2 OL iterations Jan 6 1997 00Z
Dt = 6h
Dx = 10km

9. The error after 10 OL iterations Jan 6 1997 00Z
Dt = 6h
Dx = 10km

10. The error after 40 OL iterations Jan 6 1997 00Z
Notice there is some structure in the error field
Dt = 6h
Dx = 10km

11. The error after 2 and 10 OL iterations when using a 30 minute time step (Jan 6 1997 00Z)
Dt = 30 min
Dx = 10km

12. Shear deformation (Jan 6 1997 00Z)
OL are needed to have 95% of the maximum shear
Dt = 6h
Dx = 10km

13. Outline The nonlinear convergence of existing VP models
Reasons for the slow convergence
The JFNK method
Preliminary results with the JFNK method

14. Why is the convergence so slow?
The sea-ice momentum equation is highly nonlinear
The equation is not continuously differentiable --
capping of the viscous coefficients
The linearization approach is not optimal
It is not the water drag that is responsible for the slow convergence

15. Continuously differentiable formulation for the viscous coefficients
capping (standard): z = max(P/2D,zmax) and h = ze-2
tanh (new): z = zmaxtanh(P/2Dzmax)
h=1m, A=1 D=((e112+e222)(1+e-2)+4e-2e122+2e11e22(1+e-2))1/2
capping
tanh

16. Linearization approach
Consider F(u) = z(u)u – b = 0
Taylor expansion: F(u+du) = F(u)+F’du
By requiring that F(u+du) = 0, we find
du = -F(u)/F’
where F’ = z’(u)u + z(u)
 Newton: uk = uk-1 - F(uk-1) / (z’(uk-1) uk-1 + z(uk-1) )
 Standard: uk = uk-1 - F(uk-1) / z(uk-1) uk
uk-1 X
 Standard: uk = uk-1 - F(uk-1) / z(uk-1)

17. Outline The nonlinear convergence of existing VP models
Reasons for the slow convergence
The JFNK method
Preliminary results with the JFNK method

18. We want to solve the nonlinear system of eqs:
The JFNK method
We want to solve the nonlinear system of eqs:
F(u)=A(u)u-b=0
Provide an initial guess u0
do k=1, kite
Solve J(uk-1)duk=-F(uk-1) with a Krylov method
uk = uk-1 + duk
enddo
With the tanh. Quadratic in the vicinity of the solution.
J(uk-1)v ~ ( F(uk-1+ev) - F(uk-1) ) / e

19. The preconditioned GMRES method
Krylov subspace method
low storage requirements
symmetry is not a prerequisite
parallelizable

20. Outline The nonlinear convergence of existing VP models
Reasons for the slow convergence
The JFNK method
Preliminary results with the JFNK method

21. Preliminary result at a resolution of 80 km
standard line. +
capping
standard line. +
tanh

22. Preliminary result at a resolution of 40 km
standard line. +
capping
standard line. +
tanh
Discuss problems at high resolution

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

24. Linearization approach
Consider F(u) = z(u)u – b = 0
Taylor expansion: F(u+du) = F(u)+F’du
The Newton method requires that F(u+du) = 0.
 duk = -F(uk-1) / F’(uk-1), uk = uk-1 + duk
Our example: uk = uk-1 - F(uk-1) / (z’(uk-1) uk-1 + z(uk-1) )
Standard approach: uk = b / z(uk-1)
uk = uk-1 - F(uk-1) / z(uk-1) uk
uk-1