  A new algorithm is proposed to describe the propagation of fronts advected in
the normal direction with prescribed speed function F. The assumptions on F are
that it does not depend on the front itself, but can depend on space and time.
Moreover, it can vanish and change sign. To solve this problem the Level-Set
Method [Osher, Sethian; 1988] is widely used, and the Generalized Fast Marching
Method [Carlini et al.; 2008] has recently been introduced. The novelty of our
method is that its overall computational complexity is predicted to be
comparable to that of the Fast Marching Method [Sethian; 1996], [Vladimirsky;
2006] in most instances. This latter algorithm is O(N^n log N^n) if the
computational domain comprises N^n points. Our strategy is to use it in regions
where the speed is bounded away from zero -- and switch to a different
formalism when F is approximately 0. To this end, a collection of so-called
sideways partial differential equations is introduced. Their solutions locally
describe the evolving front and depend on both space and time. The
well-posedness of those equations, as well as their geometric properties are
addressed. We then propose a convergent and stable discretization of those
PDEs. Those alternative representations are used to augment the standard Fast
Marching Method. The resulting algorithm is presented together with a thorough
discussion of its features. The accuracy of the scheme is tested when F depends
on both space and time. Each example yields an O(1/N) global truncation error.
We conclude with a discussion of the advantages and limitations of our method.
