Stability conditions for the Leapfrog‐Euler scheme with central spatial discretization of any order

In this paper a sufficient condition (Theorem 2.3) for the von Neumann stability of the Leapfrog‐Euler scheme, which uses central spatial discretization of any order for 3D convection‐diffusion equation, is derived in terms of the Courant and the diffusion numbers and the coefficients of approximation schemes. In the case of the second order differencing this condition becomes the necessary condition for the stability. Some particular sufficient conditions for the stability of the second and the fourth order schemes are also derived. A comparison of the results, which were obtained applying the derived stability conditions to compute the time step in the direct numerical simulations (DNS) of turbulent pipe flow with the help of the second and the fourth order schemes, is presented. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)