A Stable and Accurate Numerical Method to Calculate the Motion of a Sharp Interface Between Fluids

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Abstract
A spectrally accurate numerical method, which is also simple to implement, is derived to calculate the motion of a fluid interface. A numerical instability, typical of algorithms for this problem, is investigated via a linearizing approximation which shows that a trivial adjustment of the algorithm renders it numerically stable while retaining high accuracy. To test for the correct implementation of the algorithm, non-linear simultaneous equations are formulated using time derivative information only from the coded procedure, whose solutions describe progressive interfacial waves of permanent form. The behaviour of this type of algorithm when variations in the point spacing occurs along the interface is investigated and found to cause “scattering” of travelling wave modes.

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