A stability criterion of elastic wave modelling by the Fourier method

Forward modelling of wave equations by Fourier transform is widely regarded for its high precision, small memory and moderate calculation time. In forward modelling, the stability problem must be considered. This paper derives a stability criterion for the simulation of elastic wave propagation by Fourier transform. The derived stability condition is v²k²Δt² <= 4, where v is the phase velocity, k is the wavenumber and Δt is the time interval. Furthermore, stability is discussed for one-dimensional, two-dimensional and three-dimensional modelling respectively. For m-dimensional modelling by Fourier transform in elastic media, for a grid of cube, square or line segments where each side length of the grid is Δd, then, when the modelling parameters satisfy v_{\max } \Delta t/\Delta d \le 2/(\sqrt m \pi ), the modelling process is stable.