Analytical, linear stability criteria for the leap-frog, Dufort-Frankel method

Abstract

The criterion of linear numerical stability of the combined leap-frog Dufort-Frankel scheme for advective-diffusive problems in two dimensions is κ x Δx 2 + κ y Δy 2 U 2 κ x + V 2 κ y Δt 2⩽1 , where Δx and Δy are the grid spacings in the x and y directions, U and V the velocity components, κ x and κ y the diffusion coefficients, and Δt the time step. Although this stability requirement does not depend explicitly on the magnitude of the diffusivity (only on the ratio of the diffusivity coefficients), the presence of the diffusive terms renders the criterion more severe than the one obtained for purely advective problems [(| U| Δx + | V| Δy) Δt ≤ 1]. Only in one dimension are both criteria identical. Therefore, at more than one dimension, the unconditionally stable scheme (Dufort-Frankel) combined with the conditionally stable scheme (leap-frog) leads to the more restrictive stability condition. The maximum allowable time step occurs when the grid spacings in both directions have the same ratio as the square roots of the diffusivity coefficients. A new method is presented for discussing the stability of a finite-difference scheme without actually solving for the modulus of the amplification factor. This method is also extended to more general cases.