Abstract
The Leap Frog time stepping scheme (hereafter LF) partly loses its conservation properties when a Robert–Asselin filter (hereafter RA) is used to damp the computational mode. The LF + RA scheme actually leads to a well-known long term attenuation of the physical mode. Besides, the stability of the LF, e.g. the maximum permitted time step, is lowered by the use of the RA. Several methods, derived from the Laplacian approach of Marsaleix et al. (2008), are presented as an alternative to the RA. It appears that the physical mode is eventually much less impacted by higher order time filters. However, in some cases, the stability of the time stepping scheme becomes worse than that of the LF + RA. A five points scheme finally appears to preserve both the amplitude of the physical mode and the stability of the time stepping scheme. The analysis of these filters is based on a triple approach: the kinetic energy balance, the amplification factors of the oscillation equation, numerical experiments performed with a 3D circulation ocean model.
Highlights
The leapfrog time stepping scheme is a second order, three time-level, and time centred scheme that has been used for years in numerous General Circulation Models (GCMs) (Mesinger and Arakawa, 1976).There are principally two criticisms that are made about the LF scheme
The latter is generally associated to the LF scheme in order to counter the possible growth of the numerical mode permitted by the Leapfrog scheme
If the diffusion coefficient used in the M08 Laplacian filter is such that vLP 1⁄4 0:1, (2.34) says that an equivalent damping effect of the spurious 2Dt spurious oscillations is obtained with the Robert–Asselin filter provided that vRA % 0:1056
Summary
The leapfrog time stepping scheme (thereafter the LF scheme) is a second order, three time-level, and time centred scheme that has been used for years in numerous General Circulation Models (GCMs) (Mesinger and Arakawa, 1976). The second criticism, upon which the present paper focuses, concerns the Robert–Asselin time filter (Robert, 1966; Asselin, 1972) The latter is generally associated to the LF scheme in order to counter the possible growth of the numerical mode permitted by the Leapfrog scheme. The LF scheme combined to a Robert–Asselin filter is notably implemented in widespread ocean models like POM (Blumberg and Mellor, 1987) and NEMO (Leclair and Madec, 2009, 2011). The objective of the present paper is clearly to focus on what we consider to be the principal weak point of the LF scheme, namely the Robert–Asselin filter used to damp the numerical mode permitted by the LF scheme. We start to recall the differences between the Laplacian and Robert–Asselin filters
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