Extending the Courant-Friedrichs-Lewy stability condition by local eigenvalue perturbation

ABSTRACT The explicit time-marching finite-difference (FD) scheme is widely used for simulating wave propagation; however, its computational efficiency is constrained by the Courant-Friedrichs-Lewy (CFL) stability condition, which limits the maximum allowable time steps. The eigenvalue perturbation method effectively extends this condition, but it incurs prohibitive memory demand and computational cost. In contrast, the optimal spatial-filtering method requires no additional memory and achieves numerical accuracy equivalent to that of the eigenvalue perturbation method in homogeneous models. However, its performance deteriorates significantly in heterogeneous models, particularly in those with high velocity contrast. To address the excessive memory demand and computational cost of the eigenvalue perturbation method, we develop a local eigenvalue perturbation method to extend the CFL stability condition. First, we construct an augmented explicit time-marching FD scheme for the scalar wave equation; then, we derive the corresponding eigenvalue inequality relation; next, we develop a localization strategy to reduce the memory demand and computational cost of the eigenvalue perturbation method; and finally, we perform theoretical analyses and numerical experiments to demonstrate the numerical accuracy of this method. In 2D models, this method significantly reduces memory demand and computational cost by one and two orders of magnitude, respectively, compared with the eigenvalue perturbation method. Moreover, under the same numerical accuracy conditions, the maximum time step allowed by this method is approximately three times greater than that permitted by the optimal spatial-filtering method. This method enables the use of larger time steps beyond the CFL stability condition, showing promise for large-scale models, particularly those with high velocity contrast.