On the Jacobian approximation in sea ice models with viscous-plastic rheology

Abstract

Viscous-plastic rheology is widely used in the sea ice modeling community at increasingly high resolutions. Due to the high degree of non-linearity of the rheological constraints, accurate approximation of the Jacobian is required to improve the efficiency of the implicit solvers of the sea ice momentum equation in pack ice. We consider the analytical Jacobian of the ice momentum equation and assess its approximation errors in the Jacobian-free Newton–Krylov (JFNK) method and in the family of more traditional schemes which neglect the dependence of viscosity coefficients on the deformation tensor. It is shown that this dependence provides a substantial contribution to the Jacobian, especially in the regions enriched by linear kinematic features like ridges and elongated polynyas. Numerical experiments indicate that performance of the Newton solvers is also sensitive to errors associated with inexact computation of the search direction, that may be caused, in particular, by numerical approximations of the Jacobian which violate its dissipative property. Based on this analysis, an improved selective damping strategy for the Newtonian solver of the momentum equation is proposed. A series of numerical experiments conducted in simulated pack ice environment demonstrate faster convergence of the updated solver with analytical Jacobian as compared to the one based on the selectively damped JFNK method with inexact GMRES solver in the inner loop.