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2.6 Implicit time-stepping: backward method

Vertical diffusion and viscosity can be treated implicitly in time using the backward method which is an intrinsic scheme. Recently, the option to treat the vertical advection implicitly has been added, but not yet tested; therefore, the description hereafter is limited to diffusion and viscosity. For tracers, the time discretized equation is:

$\displaystyle \tau^{n+1} - \Delta t \partial_r \kappa_v \partial_r \tau^{n+1} = \tau^{n} + \Delta t G_\tau^{(n+1/2)}$ (2.25)

where $ G_\tau^{(n+1/2)}$ is the remaining explicit terms extrapolated using the Adams-Bashforth method as described above. Equation 2.25 can be split split into:
$\displaystyle \tau^*$ $\displaystyle =$ $\displaystyle \tau^{n} + \Delta t G_\tau^{(n+1/2)}$ (2.26)
$\displaystyle \tau^{n+1}$ $\displaystyle =$ $\displaystyle {\cal L}_\tau^{-1} ( \tau^* )$ (2.27)

where $ {\cal L}_\tau^{-1}$ is the inverse of the operator

$\displaystyle {\cal L}_\tau = \left[ 1 + \Delta t \partial_r \kappa_v \partial_r \right]$ (2.28)

Equation 2.26 looks exactly as 2.23 while 2.27 involves an operator or matrix inversion. By re-arranging 2.25 in this way we have cast the method as an explicit prediction step and an implicit step allowing the latter to be inserted into the over all algorithm with minimal interference.

Fig. 2.3 shows the calling sequence for stepping forward a tracer variable such as temperature.

In order to fit within the pressure method, the implicit viscosity must not alter the barotropic flow. In other words, it can only redistribute momentum in the vertical. The upshot of this is that although vertical viscosity may be backward implicit and unconditionally stable, no-slip boundary conditions may not be made implicit and are thus cast as a an explicit drag term.


next up previous contents
Next: 2.7 Synchronous time-stepping: variables Up: 2. Discretization and Algorithm Previous: 2.5 Explicit time-stepping: Adams-Bashforth   Contents
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