Maximizing the CFL number of stable time–space domain explicit finite-difference modeling

Abstract

The time–space domain explicit finite-difference (FD) method numerically solves wave equation by applying FD to approximate its spatial and temporal derivatives. This method always faces the stability problem. I theoretically analyze stability condition of this method for the acoustic wave equation, and find that the maximum Courant–Friedrichs–Lewy (CFL) number of stable modeling depends on the maximum value of the spatial FD dispersion relation. The conventional method commonly determines spatial FD coefficients by approximating the spatial FD dispersion to the exact relation within the given wavenumber area. However, outside this area, the dispersion relation and hence the maximum CFL number is uncontrollable. To make it controllable, I propose to simultaneously approximate the spatial FD dispersion relation to a given function outside this area. Further, I develop an algorithm to maximize the maximum CFL number. Stability analysis and modeling examples demonstrate that the proposed method has better stability than the conventional method, especially when the maximum effective wavenumber/frequency is small. For the 2D case, the conventional method performs stable simulation using a CFL number of no more than 0.707, but the new method can perform stable simulation using a large CFL number (for example, 1.6, 4.8, and 9.0) and non-long spatial FD stencils.