Figure 2 shows the average error as a function of grid spacing for the first and second-order solvers. Not only is the second-order formulation more accurate at large grid spacing, but its accuracy increases more rapidly as grid spacing decreases. Theory predicts the plots of average error against grid spacing to be a linear function with gradient of one for first-order methods, and two for second order methods. In practice, the fast marching results come very close to these criteria up to the limits of machine precision. Figure 2 demonstrates the superiority of the second-order fast marching formulation.
It is worth noting, at this point, that special treatment is required at the source location, since the singularity in wavefront curvature will cause numerical errors to propagate into the traveltime solution. We surround the source with a constant velocity box, within which we calculate traveltimes by ray-tracing. Errors are inversely proportional to the radius of this box. Therefore, if the radius of the box decrease with grid spacing, errors will increase linearly, reducing the accuracy of the method to first-order. For full second-order accuracy, the box size should be independent of grid spacing.