On the space discretization and time integration for level set convection

One canonical practice in the development and application of level set methods is to convect a shape represented by zero level set with a given, reversible, and periodic velocity field and test how well the original shape is recovered after each cycle. In this short letter, we mathematically show that Crank–Nicolson time integration, combined with standard Galerkin finite element, can exactly recover the original shape after each cycle, regardless of spatiotemporal resolution. This surprising finding is also numerically demonstrated by LeVeque’s problem.