The leapfrog differencing scheme is unsuitable for the representation of diffusion and damping processes. For a tendancy , representing a diffusion term or a restoring term to a tracer climatology (when present, see § 5.6), a forward time differencing scheme is used :
This is diffusive in time and conditionally stable. The conditions for stability of second and fourth order horizontal diffusion schemes are [Griffies, 2004]:
For the vertical diffusion terms, a forward time differencing scheme can be used, but usually the numerical stability condition imposes a strong constraint on the time step. Two solutions are available in NEMO to overcome the stability constraint: a forward time differencing scheme using a time splitting technique (ln_zdfexp = true) or a backward (or implicit) time differencing scheme (ln_zdfexp = false). In , the master time step t is cut into fractional time steps so that the stability criterion is reduced by a factor of . The computation is performed as follows:
This scheme is rather time consuming since it requires a matrix inversion, but it becomes attractive since a value of 3 or more is needed for N in the forward time differencing scheme. For example, the finite difference approximation of the temperature equation is:
(3.8) is a linear system of equations with an associated matrix which is tridiagonal. Moreover, and are positive and the diagonal term is greater than the sum of the two extra-diagonal terms, therefore a special adaptation of the Gauss elimination procedure is used to find the solution (see for example Richtmyer and Morton ).
Gurvan Madec and the NEMO Team
NEMO European Consortium2017-02-17