A linear algorithm for solving non-linear isothermal ice-shelf equations

Abstract. A linear non-iterative algorithm is suggested for solving nonlinear isothermal steady-state Morland–MacAyeal ice shelf equations. The idea of the algorithm is in replacing the problem of solving the non-linear second order differential equations for velocities with a system of linear first order differential equations for stresses. The resulting system of linear equations can be solved numerically with direct methods which are faster than iterative methods for solving corresponding non-linear equations. The suggested algorithm is applicable if the boundary conditions for stresses can be specified. The efficiency of the linear algorithm is demonstrated for one-dimensional and two-dimensional ice shelf equations by comparing the linear algorithm and the traditional iterative algorithm on derived manufactured solutions. The linear algorithm is shown to be as accurate as the traditional iterative algorithm but significantly faster. The method may be valuable as the way to increase the efficiency of complex ice sheet models a part of which requires solving the ice shelf model as well as to solve efficiently two-dimensional ice-shelf equations.