Accurate interface-tracking for arbitrary Lagrangian-Eulerian schemes

preprint numerics no. 3/2008 norwegian university of science and technology trondheim, norway We present a new method for tracking an interface immersed in a given velocity field. The method 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 maintain a good point distribution for the grid points along the interface. The method combines information from a pure Lagrangian approach with information from an ALE approach. No integration backwards in time is needed; instead, the new method relies of 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 been able to verify first, second, and third order temporal accuracy for the new method by solving two-dimensional model problems with known solutions.