Time-space-domain mesh-free finite difference based on least squares for 2D acoustic-wave modeling

摘要

Finite-difference (FD) methods approximate derivatives through a weighted summation of function values from neighboring nodes. Traditionally, these neighboring nodes are assumed to be distributed regularly, such as in square or rectangular lattices. To improve geometric flexibility, one option is to develop FD in a mesh-free discretization, in which scattered nodes can be placed suitably with respect to irregular boundaries or arbitrarily shaped anomalies without coordinate transformation or forming any triangles or tetrahedra, etc. These mesh-free FDs have had successful applications, especially in computational geoscience. However, they are all space-domain FD schemes, in which FD coefficients are derived by approximating spatial derivatives individually in the space domain. For acoustic-wave modeling, it has been proven that spacedomain FD methods normally have higher dispersion error than time-space-domain FD methods, which determine FD coefficients by approximating the time-space-domain dispersion relation. Now, we have developed a time-space-domain mesh-free FD based on minimizing the absolute error of the dispersion relation by least-squares (LS) for 2D acoustic-wave modeling. The matrix used to solve for FD coefficients in our method is determined by the spatial distribution of the nodes in a local FD stencil, whereas the temporal step size and velocity information are considered in the right side of the linear system. This feature of considering both spatial and temporal effects allows our proposed mesh-free LS-based FD to obtain greater temporal accuracy adaptive to different Courant-Friedrichs-Lewy parameters than pure space-domain mesh-free FDs. Under several 2D acoustic scenarios, the advantage was proven by comparing our method with radial-basis-function-generated FD, which is one of the most popular mesh-free FDs and has been applied in elastic wave modeling.