The response of glaciers and ice sheets to climate or sea-level forcing over a range of time scales must be calculated for several different climate variation studies. Most time-dependent ice-flow modelling uses finite differences to solve the continuity equation. Any numerical scheme can be treated as a filter f(x) whose input is the ice profile at time step n and whose output is the ice profile at time step (n+1). The physics of ice deformation is contained in f(x). Employing the usual linearizations (e.g. Reference PatersonPaterson 1981: 254), the filter f(x) has a complex transfer function F(k) in the wave-number domain. The standard linear stability analysis is obtained by choosing time and space mesh intervals so that the modulus of F(k) is never greater than unity. In addition, the modulus of F(k) describes the rate of diffusion of wave disturbances, and the phase of F(k) describes the propagation speed. The transfer functions for several finite-difference schemes are compared to an analytical solution. For given time and space mesh intervals, implicit schemes are more accurate than explicit schemes. Schemes with a staggered grid for the flux calculations can model diffusion out to the highest wave number seen by the mesh. Non-staggered schemes do not attenuate these wave numbers; they require additional numerical smoothing to suppress nonlinear aliasing effects which arise at the high wave numbers regardless of the time step sizes. This smoothing must be restricted to wave numbers higher than those contributing to the ice-profile spectrum.

Large errors in amplitude and propagation speed can result even at low wave numbers from using time steps larger than the limit set by standard stability analysis, then smoothing out the resulting instability. In this case, the filter f(x) based on the physics of deforming ice has been effectively replaced by an arbitrary low-pass smoothing filter with no physical significance.