A Low Complexity Algorithm for Non-Monotonically Evolving Fronts

Abstract

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. The novelty of our method is that its overall computational complexity is predicted to be comparable to that of the Fast Marching Method (Sethian in Proceedings of the National Academy Sciences 93:1591---1595, 1996); (Vladimirsky in Interfaces Free Bound 8(3):281---300, 2006) in most instances. This latter algorithm is $$\mathcal {O}(N^{n}\log N^{n})$$O(NnlogNn) if the computational domain comprises $$N^{n}$$Nn points. We use it in regions where the speed is bounded away from zero--and switch to a different formalism when $$F \approx 0$$Fź0. 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 and geometric properties of those equations are addressed. We then propose a convergent and stable discretization of the PDEs. 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 $$\mathcal {O}(1/N)$$O(1/N) global truncation error. We conclude with a discussion of the advantages and limitations of our method.