Recent papers by Shchepetkin (2015) and Lemarié et al. (2015) have emphasized that the time-step of an oceanic model with an Eulerian vertical coordinate and an explicit time-stepping scheme is very often restricted by vertical advection in a few hot spots (i.e. most of the grid points are integrated with small Courant numbers except just few spots). The consequence is that the numerics for vertical advection must have good stability properties while being robust to changes in Courant number in terms of accuracy. An other constraint is the strict control of numerical mixing imposed by the highly adiabatic nature of the oceanic interior. The same applies for remapping schemes when ALE coordinates are used. We propose to examine in this talk the possibility of mitigating vertical CFL restriction, while avoiding numerical inaccuracies associated with standard implicit schemes (i.e. large sensitivity of the solution on Courant number, large phase delay, and possibly excess of numerical damping). Several regional oceanic models have been successfully using fourth order compact spatial discretizations for vertical advection. In this talk we present a space-time generalization of compact schemes. In particular, we derive a generic expression for a fourth-order (one-step) coupled time and space compact scheme (see Daru & Tenaud (2004) for a thorough description of coupled time and space schemes). Among other properties, we show that this scheme is non dissipative, unconditionally stable, and has very good accuracy properties especially for Courant numbers smaller than 1 while having a very small computational cost. Furthermore, we show how this scheme can be made monotonic without compromising its stability properties. We emphasize that the scheme can be successfully used in number of different situations:

• as an unconditionally stable vertical advection scheme in an oceanic model with quasi-Eulerian vertical coordinate (provided a degradation of time accuracy for large Courant numbers)
• as a scheme to compute the so-called ”fractional flux” in a flux-form semi-Lagrangian scheme à la Lin & Rood (1996) to provide a stable scheme even for large Courant numbers
• as a remapping scheme in an ALE framework following Dukowicz & Baumgardner (2000) We illustrate the properties of the scheme and compare it to existing fourth-order accurate in time and space schemes using linear and nonlinear numerical experiments.