13.5.1 First Order Space Convex

The following discretization is a first order finite difference scheme for convex speed functions [87]. This and the scheme in Section 13.5.2 converge to the correct viscosity solution [26,88]. $\phi_{ijk}^n$ are the grid values of the level set function $u$ at time step $n$ and $F_{ijk}$ are the values of the speed function. Here $D_{ijk}^{+x}$ is written shortly for $D^{+x} \phi_{ijk}^n$, which is the forward difference in $x$ direction. The explicit values for the next time step are

$$\phi_{ijk}^{n+1} = \phi_{ijk}^n - \Delta t \bigl( \max(F_{ijk},0) \nabla^+ + \min(F_{ijk},0) \nabla^- \bigr),$$

where

and