the liquid and is the surface tension. This choice of timescale makes the scaled surface tension equal to 1; i.e., the A system of partial differential equations that approximate the governing equations for inviscid free surface flow subject to surface Weber number is set to one. Henceforth, only the scaled tension is presented. The approximation is based on linearizing the problem will be considered. Let x and y be the Cartesian velocity together with a small scale approximation of the perturbacoordinates in the cross-sectional plane of the jet, and let tion of the velocity. Two Dirichlet problems must be solved to form t be the time. We assume that the liquid occupies the simply the approximate system, after which it can be evolved without solvconnected time-dependent domain ⍀(t) with a smooth ing Dirichlet problems. The accuracy of the solution is determined by how often the velocity term is linearized. This time-interval is boundary ⌫(t). It is convenient to describe the motion in called ⌬T. We show that the error in the solution of the approximate the positively oriented Lagrangian coordinate 0 Յ Ͱ Յ 2ȏ, system at a fixed time T is of the order O (⌬T 2 ). We demonstrate such that the boundary at time t Ն 0 is given by x ϭ X(Ͱ, numerically that the error is closely correlated to the size of the t), y ϭ Y(Ͱ, t) and the velocity potential on the boundary normal velocity and that there is a stability limit of the form ⌬T Յ is ϭ (Ͱ, t). The governing equations for , X, Y on C/(͉u n ͉ ȍ ) Ͳ , where u n denotes the normal velocity of the free surface and Ͳ Ȃ 2.6. Importantly, C is independent of the resolution, so the ⌫(t) are time-step ⌬T can be chosen independently of the number of grid points, N. This is in contrast to the original system, where the stabil-
ity limit of the time-step is ⌬t Յ O (N Ϫ3/2 ) and a fixed number of Dirichlet problems have to be solved per time-step. By numerical
experiments, we demonstrate that the approximate system requires less than 10% of the CPU time used by the original system to solve
the problem very accurately.