Improved convergence and stability properties achieved in a three-dimensional higher-order ice sheet model

We present a finite difference implementation of a three-dimensional higher-order ice sheet model. In com- parison to a conventional centred difference discretisation it enhances both numerical stability and convergence. In or- der to achieve these benefits the discretisation of the gov- erning force balance equation makes extensive use of infor- mation on staggered grid points. Using the same iterative solver, a centred difference discretisation that operates exclu- sively on the regular grid serves as a reference. The reprise of the ISMIP-HOM experiments indicates that both discreti- sations are capable of reproducing the higher-order model inter-comparison results. This setup allows a direct compar- ison of the two numerical implementations also with respect to their convergence behaviour. First and foremost, the new finite difference scheme facilitates convergence by a factor of up to 7 and 2.6 in average. In addition to this decrease in computational costs, the accuracy for the resultant veloc- ity field can be chosen higher in the novel finite difference implementation. Changing the discretisation also prevents build-up of local field irregularites that occasionally cause divergence of the solution for the reference discretisation. The improved behaviour makes the new discretisation more reliable for extensive application to real ice geome- tries. Higher accuracy and robust numerics are crucial in time dependent applications since numerical oscillations in the velocity field of subsequent time steps are attenuated and divergence of the solution is prevented.