A Survey of Finite-difference Schemes for the Primitive Equations for a Barotropic Fluid

Ten different finite-difference schemes for the numerical integration of the primitive equations for the freesurface model are tested for stability and accuracy. The integrations show that the quadratic conservative and the total energy conservative schemes are more stable than the usual second-order conservative scheme. But the most stable schemes are those in which the finite-difference approximations to the advection terms are calculated over nine grid points in space and therefore contain a form of smoothing, and. the generalized Arakawa scheme, which for nondivergent flow conserve mean vorticity, mean kinetic energy, and mean square vorticity. If the integrations are performed for more than 3 days, it is shown that more than 15 grid points per wavelength are probably needed to describe with accuracy the movement and development of the shortest wave that initially is carrying a significant part of the energy. This is true even if a fourth-order scheme in space is used. Long-term integrations using the leapfrog method or midpoint rule in time may lead to instability of the integration from the increase of energy on the computational modes. Elimination of the computational modes by using a smoothing operator in time or by using other multistep time-integration methods, which damp the computational modes, will improve the stability of the integration. As a rule, linear stable one-step methods have strong built-in dissipation and in a few days will damp out most of the initial perturbation energy, even if they are used only intermittently once a day.