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.

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

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

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.

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.

The nonlinear numerical convergence Jan 6 1997 00Z Dt = 6h Dx = 10km FC solution in 10500 OL. Limited by machine precision

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

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

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

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

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

Shear deformation (Jan 6 1997 00Z) 100-500 OL are needed to have 95% of the maximum shear Dt = 6h Dx = 10km

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

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

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

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)

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

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

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

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

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

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

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

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