the curve moves in the normal direction with a speed , then the level-set function satisfies the level-set equation Here, is the Euclidean norm (denoted customarily by single bars in PDEs), and is time. This is a partial differential equation, in particular a Hamilton–Jacobi equation, and can be solved numerically, for example, by using finite differences on a Cartesian grid. The numerical solution of the level-set equation, however, requires sophisticated techniques. Simple finite-difference methods fail quickly. Upwinding methods, such as the Godunov method, fare better; however, the level-set method does not guarantee the conservation of the volume and the shape of the level set in an advection field that does conserve the shape and size, for example, uniform or rotational velocity field. Instead, the shape of the level set may get severely distorted, and the level set may vanish over several time steps. For this reason, high-order finite-difference schemes are generally required, such as high-order essentially non-oscillatory (ENO) schemes, and even then the feasibility of long-time simulations is questionable. Further sophisticated methods to deal with this difficulty have been developed, e.g., combinations of the level-set method with tracing marker particles advected by the velocity field. Example Consider a unit circle in , shrinking in on itself at a constant rate, i.e. each point on the boundary of the circle moves along its inwards pointing normal at some fixed speed. The circle will shrink and eventually collapse down to a point. If an initial distance field is constructed (i.e. a function whose value is the signed euclidean distance to the boundary, positive interior, negative exterior) on the initial circle, the normalised gradient of this field will be the circle normal. If the field has a constant value subtracted from it in time, the zero level (which was the initial boundary) of the new fields will also be circular and will similarly collapse to a point. This is due to this being effectively the temporal integration of the Eikonal equation with a fixed front velocity. In combustion, this method is used to describe the instantaneous flame surface, known as the G equation. History The level-set method was developed in 1979 by Alain Dervieux, and subsequently popularized by Stanley Osher and James Sethian. It has become popular in many disciplines, such as image processing, computer graphics, computational geometry, optimization, computational fluid dynamics, and computational biology. A number of level-set data structures have been