Detecting contour crossings in contour dynamical and contour-advective semi-Lagrangian simulations

Contour dynamics and contour-advective methods are commonly used numerical techniques for simulating inviscid fluid motions. In these methods the vorticity or potential vorticity of a flow is represented by a series of contours which are advected according to the prevailing velocity field. In some circumstances the contours may cross, eroding the accuracy of the numerical solution and violating the equations of motion. This paper describes an automated method for explicitly revealing such crossings, first considering the case of determining if two contours cross and then later the more general case of determining if and where an arbitrary number of contours cross.