A comparison of the accuracy and stability of numerical schemes used to solve the equations derived from the shallow-shelf approximation

Abstract. The shallow-shelf approximation (SSA) is widely used to model the evolution of fast-flowing ice that is either floating or sliding rapidly over the bed, resulting in plug flow. However, so far there has been no rigorous, comparative study of the different numerical approaches to solve the SSA equations. We calculate the conditions for different numerical schemes to remain stable and accurate when solving the linear 1D SSA equations for a simple geometry. The comparison spans central difference and upwind schemes, explicit and semi-implicit multistep schemes, and explicit or implicit monolithic schemes. Also considered are schemes time stepping using the Θ-method, where the value of Θ is used to form a weighted combination of the forward and backward Euler methods. We find that monolithic schemes remain stable at all tested time steps for Θ ≥ 0.5 whilst multistep schemes become unstable at large time steps. Of the tested schemes the monolithic finite element scheme with Θ = 0.5 remains accurate for the largest range of time steps.