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

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 ...

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

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

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.

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

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

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

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

The reference solution after 11 days

Newton iterations and CPU time as a function of Δt

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

RMSE (thickness) after 11 days (10km)

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

Thank you!!!

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

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

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

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

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

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