Editing Level-set method (section)

==The level-set equation==
If the curve <math>\Gamma</math> moves in the normal direction with a speed <math>v</math>, then by chain rule and implicit differentiation, it can be determined that the level-set function <math>\varphi</math> satisfies the ''level-set equation'' 
:<math>\frac{\partial\varphi}{\partial t} = v|\nabla \varphi|.</math>
Here, <math>|\cdot|</math> is the [[Euclidean norm]] (denoted customarily by single bars in partial differential equations), and <math>t</math> is time. This is a [[partial differential equation]], in particular a [[Hamilton–Jacobi equation]], and can be solved numerically, for example, by using [[finite difference]]s on a Cartesian grid.<ref name=osher>{{cite book |last=Osher |first=Stanley J. |authorlink = Stanley Osher |author2=Fedkiw, Ronald P. |authorlink2=Ronald Fedkiw |title=Level Set Methods and Dynamic Implicit Surfaces|publisher=[[Springer-Verlag]] |year=2002 |isbn= 978-0-387-95482-0}}</ref><ref name=sethian>{{cite book |last=Sethian |first=James A. |authorlink = James Sethian |title= Level Set Methods and Fast Marching Methods : Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science|publisher=[[Cambridge University Press]] |year=1999 |isbn= 978-0-521-64557-7}}</ref> 

However, the numerical solution of the level set equation may require advanced techniques. Simple finite difference methods fail quickly. [[Upwind scheme|Upwinding]] methods such as the [[Godunov method]] are considered better; however, the level set method does not guarantee preservation of the volume and shape of the set level in an advection field that maintains shape and size, for example, a uniform or [[rotational velocity]] field. Instead, the shape of the level set may become distorted, and the level set may disappear over a few time steps. Therefore, high-order finite difference schemes, such as high-order essentially non-oscillatory (ENO) schemes, are often required, and even then, the feasibility of long-term simulations is questionable. More advanced methods have been developed to overcome this; for example, combinations of the leveling method with tracking marker particles suggested by the velocity field.<ref>{{Citation |last1 = Enright |first1 = D. |last2 = Fedkiw |first2 = R. P.| last3 = Ferziger |first3 = J. H. |authorlink3 = Joel H. Ferziger| last4 = Mitchell |first4 = I.| title = A hybrid particle level set method for improved interface capturing| journal = J. Comput. Phys.| volume = 183 |issue = 1 |year = 2002 |pages = 83&ndash;116| url = http://www.cs.ubc.ca/~mitchell/Papers/myJCP02.pdf |doi=10.1006/jcph.2002.7166|bibcode = 2002JCoPh.183...83E |citeseerx = 10.1.1.15.910}}</ref>