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We further study the properties of the back and forth error compensation and correction (BFECC) method for advection equations such as those related to the level set method and for solving Hamilton-Jacobi equations on unstructured meshes. In particular, we develop a new limiting strategy which requires another backward advection in time so that overshoots/undershoots on the new time level get exposed when they are transformed back to compare with the solution on the old time level. This newdoi:10.1090/mcom/3026 fatcat:uoaqcmcfurc4nmd6kqjpzx5leq