A propagating interface can develop corners and discontinuities
as it advances. For the computation of first-arrival traveltimes
of the eikonal equation, we first consider the level set method
called the group marching method (GMM), which is an optimal
variant of the fast marching method of Tsitsiklis and Sethian.
Then we consider a second-order ENO scheme implemented as an
expanding-box scheme, which can be easily instable for turning
rays. To overcome the instability of such an expanding box
scheme, the algorithm incorporates a dynamic Down-'N'-Out (DNO)
marching, followed by correction-by-iteration sweeps called the
Post Sweeping (PS). The resulting algorithm (ENO-DNO-PS) requires
O(N) operations in practice and has been successfully applied to
the computation of both isotropic and anisotropic traveltimes in
heterogeneous media.
Accuracy and efficiency would be compared for the optimal algorithms.
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