A new numerical approach to solve 1D Viscous Plastic Sea Ice Momentum Equation

While there has been a colossal effort in ongoing decades, the ability to simulate ocean ice has fallen behind various parts of the climate system and most Earth System Models are unable to capture the observed adversities of Arctic sea ice, which is, as it were, attributed to our frailty to determine sea ice dynamics. Viscous Plastic rheology is the most by and large recognized model for sea ice dynamics and it is expressed as a set of partial differential equations that are hard to tackle numerically. Using the 1D sea ice momentum equation as a prototype, we use the method of lines based on Euler's backward method. This results in a nonlinear PDE in space only. At that point, we apply the Damped Newton’s method which has been introduced in Looper and Rapetti et al. [2] and used and generalized to 2D in Saumier et al. [1] to solve the Monge-Ampere equation. However, in our case, we need to solve 2nd order linear equation with discontinuous coefficients during Newton iteration. To overcome this difficulty, we use the Finite element method to solve the linear PDE at each Newton iteration. In this paper, we will show that with the proper choice of a damping factor, convergence can be guaranteed and the numerical solution indeed converges efficiently to the continuum solution unlike other numerical approaches that typically solve an alternate set of equations and avoid the difficulty of the Newton method for a large nonlinear algebraic system. The PDE approach is more stable.

[1] L.P. Saumier,B.Khouider. An Efficient Numerical Algorithm for the L2 Optimal Transport Problem with Applications to Image Processing,University of Victoria, 2010.

[2] G.Loeper and F.Rapetti. Numerical solution of the Monge-Amp‘ere equation by a Newton’s algorithm. C. R. Acad. Sci. Paris, I(340):319–324, 2005.