I am solving the Eikonal Equation in 2D:
$ | \nabla T(x,y)|=1/V(x,y) $
for the traveltime, T(x,y), from a starting point: $ T(x_0,y_0) = 0$.
The curves $ T(x,y)=C$ forms closed contours around the starting point.
However the Eikonal equation do not impose any smoothness requirements on the contours.
As an example consider the following case:
Figure: Black circle: starting point, color: velocity, V(x,y), red=fast, blue=slow.
The contours would have no problem getting trough the narrow red "gap".
The way I look at this: the Eikonal equation models sound, and sound have no problems getting trough narrow spaces.
I would like instead to model something more "viscous" so that the contours were smooth and could NOT get trough such a narrow gap.
What would be the PDE for that?
And how would I solve it numerically? Today I am using the Fast Marching Method. Would it possible to adapt this method to the new equation?