Combining the VP and EVP models to solve

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 1

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

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

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

The nonlinear system of equations 5

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

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

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

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

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

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

Computational efficiency 12

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

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

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 212 -998 -3331 if you know what’s wrong!!! 15

Thank you! 16

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Viscous-plastic formulation q . -P/2 -p ridge lead Hibler, 1979 18

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 (6 h) - climatological currents 19

Computational gain of JFNK over the standard solver Quality of approx. solution 20

Typical shear deformation field (10 km) 21

Failures of the two solvers 22

In matrix form… 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))-1 F(uk-1)

Failure of the line search method 24