Accurate interface-tracking for arbitrary Lagrangian–Eulerian schemes

We present a new method for tracking an interface immersed in a given velocity field which is particularly relevant to the simulation of unsteady free surface problems using the arbitrary Lagrangian–Eulerian (ALE) framework. The new method has been constructed with two goals in mind: (i) to be able to accurately follow the interface; and (ii) to automatically achieve a good distribution of the grid points along the interface. In order to achieve these goals, information from a pure Lagrangian approach is combined with information from an ALE approach. Our implementation relies on the solution of several pure convection problems along the interface in order to obtain the relevant information. The new method offers flexibility in terms of how an “optimal” point distribution should be defined. We have proposed several model problems, each with a prescribed time-dependent velocity field and starting with a prescribed interface; these problems should be useful in order to validate the accuracy of interface-tracking algorithms, e.g., as part of an ALE solver for free surface flows. We have been able to verify first, second, and third order temporal accuracy for the new method by solving these two-dimensional model problems.