An optimal spatial-filtering method derived from eigenvalue perturbation for extending the Courant-Friedrichs-Lewy stability limit

Abstract

Explicit time-marching schemes are widely used in numerical simulations. However, their maximum time step is constrained by the Courant-Friedrichs-Lewy (CFL) stability limit. Two methods have been developed to extend this limit: (1) the eigenvalue perturbation method, which exhibits high numerical accuracy but incurs unaffordable memory demand and computational costs, even for middle-scale models, and (2) the spatial-filtering method, which can be implemented easily but results in significant numerical errors under large time steps. However, the intrinsic relationship between these two methods remains unknown. We reveal the intrinsic relationship between these two methods by considering the eigenvalue perturbation method for a homogeneous model. Using this relationship, we derive an analytical spatial filter for the homogeneous model and develop an optimal spatial filter for heterogeneous models. Compared to the classical spatial-filtering method, which removes all high-wavenumber components, our method retains all high-wavenumber components that contribute to wave propagation while eliminating other high-wavenumber components that cause instability. For the same numerical accuracy, the maximum time step allowed by the proposed method is approximately twice that of the classical spatial-filtering method. Compared to the eigenvalue perturbation method, our method can be used in large-scale models without additional memory consumption and computational cost. This can significantly accelerate the pseudospectral method with significantly larger time steps beyond the CFL stability limit, which is particularly promising for large-scale models with a fine grid.