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2.4 Numerical Schemes for Barotropic Models     continued...

The implicit treatment (to be defined below) of the barotropic equations invokes the inversion of matrices via direct or iterative solvers. It is precisely in this area that most of the recent advances in the numerical solution of partial differential equations have taken place. Some of these advances have changed our concepts as to which iterative or direct matrix solvers are ``fastest'', because ``fastness'' depends nowadays on compatibility with computer hardware as well as on numerical convergence rates! This has become very obvious since the introduction of vector pipeline computers, and even more so with the appearance of massively parallel ones.

At the same time, the different nature of new computer architectures and the increasing size of computer memories are leading to renewed consideration of the explicit treatment of the barotropic mode. After all, as the number of vertical levels or layers increases (exceeds 20, 30 or even 60), or as we add additional transport equations for the various biochemical species, the fraction of computer time spent in solving the 2-D barotropic equations per level tends to decrease. At the same time, we notice that most explicit techniques are usually fully vectorizable and ``parallelizable'', thus giving strong competition to the implicit techniques on computers of this and the next decade.