A wide variety of problems can be formulated as an interface propagation. Some examples are burning flames, waves in water and physical boundaries. The fast marching methods and narrow band level set method are useful for finding a solution to these problems. The fast marching method is associated with the boundary value problem, and as such can only be used for a propagation which strictly expands or contracts. It is in contrast with the narrow band level set method, which is associated with the initial value formulation; it can be used for the propagation of interfaces which both expand and contract. There is a parallel of both of these problems to general wave equations. Thus, by solving the Hamilton-Jacobi equation with an appropriate flux function, numerical approximation schemes for both propagation methods can be naturally formulated in a way such that the correct solution is obtained. The final formulation of the two algorithms proves to be both robust and efficient, although they do not produce a very accurate solution using first order approximations.