Accurate interface-tracking of surfaces in three dimensions for arbitrary Lagrangian–Eulerian schemes

We extend the computational method presented in [1] for tracking an interface immersed in a given velocity field to three spatial dimensions. The proposed method is particularly relevant to the simulation of unsteady free surface problems using the arbitrary Lagrangian–Eulerian framework, and has been constructed with two goals in mind: (i) to be able to accurately follow the interface; and (ii) to automatically maintain a good distribution of the grid points along the interface. The method combines information from a pure Lagrangian approach with information from an ALE approach. The new method offers flexibility in terms of how an “optimal” point distribution should be defined, and relies on the solution of two-dimensional surface convection problems. We verify the new method by solving model problems both in the single and multiple spectral element case, and we compare this method with other traditional alternatives. We have been able to verify first, second, and third order temporal accuracy for the new method by solving these three-dimensional model problems.