Convergence and accuracy of sea ice dynamics solvers. Martin Losch (AWI) Annika Fuchs (AWI), Jean-François Lemieux (EC)

Transcription

1 Convergence and accuracy of sea ice dynamics solvers Martin Losch (AWI) Annika Fuchs (AWI), Jean-François Lemieux (EC)

2 Sea ice thickness with MITgcm (JPL)

3 Review of 2D momentum equations m Du Dt = mf k u + air + ocean m (0)+F, (1) F j = i ij = divergence of symmetric stress tensor of rank 2 Viscous-Plastic (VP) constitutive law (rheology): ij = 2 ij +[ ] kk ij P 2 ij. (2) EVP equations (NOT intended to be a different rheology): 1 E ij t ij + 4 kk ij + P 4 ij = ij. (3) ij = 1 2 ( iu j + j u i )=strain rates P = P hc e C (1 c) =min P 2, max = e 2

4 Review of 2D momentum equations m Du Dt = mf k u + air + ocean m (0)+F, (1) F j = i ij = divergence of symmetric stress tensor of rank 2 Viscous-Plastic (VP) constitutive law (rheology): ij = 2 ij +[ ] kk ij P 2 ij. (2) EVP equations (NOT intended to be a different rheology): 1 E ij t ij + 4 kk ij + P 4 ij = ij. (3) ij = 1 2 ( iu j + j u i )=strain rates P = P hc e C (1 c) =min P 2, max = e 2

5 Review of 2D momentum equations m Du Dt = mf k u + air + ocean m (0)+F, (1) F j = i ij = divergence of symmetric stress tensor of rank 2 Viscous-Plastic (VP) constitutive law (rheology): ij = 2 ij +[ ] kk ij P 2 ij. (2) EVP equations (NOT intended to be a different rheology): 1 E ij t ij + 4 kk ij + P 4 ij = ij. (3) ij = 1 2 ( iu j + j u i )=strain rates P = P hc e C (1 c) =min P 2, max = e 2

6 classical implicit VP-solver: the Picard solver for A(x) x = b(x) A(x k-1 ) x k-1 b(x k-1 ) = F(x k-1 ) outer (non-linear) loop: until (some criterion), solve A(x k-1 ) x k = b(x k-1 ) (x k ) 2 F(x k ) Lemieux and Tremblay (2009)

7 classical implicit VP-solver: the Picard solver for A(x) x = b(x) A(x k-1 ) x k-1 b(x k-1 ) = F(x k-1 ) outer (non-linear) loop: until (some criterion), solve A(x k-1 ) x k = b(x k-1 ) MITgcm default (x k ) 2 F(x k ) Lemieux and Tremblay (2009)

8 2 OL full converged 20 cm/s differences 5 cm/s 10 OL 40 OL 2 cm/s 0.5 cm/s Lemieux and Tremblay (2009) (for long time steps)

9 EVP isn t any better (but faster) reference EVP, 120 sub-cycles EVP, 1980 sub-cycles Lemieux et al. (2012), shear and divergence ¼ (per day)

10 Author's personal copy EVP isn t any better (but faster) Author's personal copy 5940 et al. / Journal of231 Computational J.-F. Lemieux J.-F. et al. /Lemieux Journal of Computational Physics (2012) Physics 231 (2012) reference Author's personal copy EVP, 120 sub-cycles Fig. 9. Divergence Greenland as simulated by the EVP with N sub ¼ (a) and with N sub ¼ 1920 (b) on 18 January Z. The advective time ste J.-F. Lemieux et al. / Journal of Computational Physics 231 north (2012)of "1 is 20 min. To see the details, the divergence is capped to ±0.025 day EVP, N =120 EVP, N =1920 sub sub JFNK, γ =10 3 nl as simulated by the EVP with N sub gence is capped to ±0.025 day"1. 3 PDF EVP, 1980 sub-cycles Z. The advective time step"06 ¼ 120 (a) with ¼ 1920 (b) 18 January sub divergence Fig.and 8. Shear (a)nand (b) aton 10-km resolution 2002 obtained with the Picard solver with c ¼ 10 Lemieux et al. (2012), shear and divergence (per day) and advective time step of 10 s (the reference nl solution) on 18 January Z. Shear (c) and divergence (d) obtained with the EVP with 120 subcycles. Shear (e) and divergence (f) obtained with the EVP with N sub ¼ 1920 on 18 January Z. The advective time step for the EVP solver is 20 min. For clarity, the shear is capped to 0.2 day"1 and the 6 divergence to ±0.05 day"

11 new to sea ice dynamics: the JFNK solver [A(x k-1 ) x k b(x k-1 ) = F(x k )] (0 =) F(x k-1 + x k ) = F(x k-1 ) + F (x k-1 ) x k, F = J(acobian) until F(x k ) < tol, solve J(x k-1 ) x k = F(x k-1 ) and update x k = x k-1 + x k F(x) F(x k ) Stand cap Stand tanh JFNK 10 1 residual norm x 3 x 2 x 1 x 0 Newton method x iteration non-linear iterations Lemieux et al. (2010)

12 JFNK in the MITgcm FGMRES with preconditioner (LSR, Zhang and Hibler, 1997) exact Jacobian times vector possible by AD parallel code: - scalar products in FGMRES - restricted additive Schwarz (RAS) method in LSR vector code: - iterative preconditioner (LSR) - coloring (zebra) method Losch et al. (2014)

13 Parallel performance of solvers (4km Arc3c configura3on) speed up

14 Does it matter? after nearly 40 years of simulation: average of Oct, 1995 with 2 OL iterations with F <10-4

15 Timing of solvers (Is JFNK really faster?)

16 Accurate solvers affect ice thickness distribution linear kinematic features computer time!!!

17 Summary traditional Picard solver converges slowly EVP solver does not converge at all (to VP) JFNK-solver is efficient but expensive

18 Summary traditional Picard solver converges slowly EVP solver does not converge at all (to VP) JFNK-solver is efficient but expensive JFNK-solver is available for large-scale problems