On Accuracy and EÆciency of 3D Eikonal Solvers

The article is concerned with accuracy and eÆciency issues for the numerical solution of the eikonal equation in heterogeneous media in three-dimensional space. A new second-order nite di erence scheme is introduced to approximate the traveltime derivatives exactly for elliptical wavefronts. Its ENO-variant, called the Quadratic ENO (QENO), is applied for the traveltime computation in realistic media. QENO shows an excellent accuracy, in particular, near the point source where the wavefront curvature is large. For stability, the expanding-box method requires to adopt an iterative procedure of which each iteration incorporates various directional marchings. Accurate initialization and the introduction of substeps in the pseudo-temporal direction turn out to be two important ingredients for a fast convergence of the iteration. With properly chosen algorithm parameters, the iterative algorithm converges in 2-3 iterations independently on the problem size in most realistic media.