We introduce a new level set method for motion in normal direction. It is based on a formulation in the form of a second order forward-backward diffusion equation. The equation is discretized by the finite volume method. We propose a semi-implicit time discretization taking into account the forward diffusion part of the solution in an implicit way, while the backward diffusion part is treated explicitly. When forward diffusion dominates, a straightforward reconstruction of the solution is used, while larger (smoothing) stencils are used when backward diffusion dominates. The method is precise on coarse grids and is second order accurate for smooth solutions. Numerical experiments show an optimal coupling of time and space steps with $tau=h$, and no stronger CFL condition is required. Numerical tests with the scheme are discussed on representative examples.

@Article{MO10,
  author       = {Mikula, K. and Ohlberger, M.},
  title        = {A New Level Set Method for Motion in Normal Direction Based on a Semi-Implicit Forward-Backward Diffusion Approach},
  journal      = {SIAM J. Sci. Comput. },
  number       = {3},
  volume       = {32},
  pages        = {1527--1544 },
  year         = {2010},
  doi          = {10.1137/09075946X},
  url          = \{/2010/MO10},
}