  We present compact semi-implicit finite difference schemes on structured
grids for numerical solutions of the advection by an external velocity and by a
speed in normal direction that are applicable in level set methods. The most
involved numerical scheme is third order accurate for the linear advection with
a space dependent velocity and unconditionally stable in the sense of von
Neumann stability analysis. We also present a simple high-resolution scheme
that gives a TVD (Total Variation Diminishing) approximation of the spatial
derivative for the advected level set function. In the case of nonlinear
advection, the semi-implicit discretization is proposed to linearize the
problem. The compact form of implicit stencil in numerical schemes containing
unknowns only in the upwind direction allows applications of efficient
algebraic solvers like fast sweeping methods. Numerical tests to evolve a
smooth and non-smooth interface and an example with a large variation of
velocity confirm the good accuracy of the methods and fast convergence of the
algebraic solver even in the case of very large Courant numbers.
