Search

We start with the basic equations for a compressible, non-hydrostatic flow and scale them based on oceanographic data to obtain a set of hyperbolic equations that is well-posed for the initial boundary value problem. We find a distinct advantage in using salinity and potential temperature as primitive variables instead of density, since the equations that we use for a compressible flow are then linearly independent. We scale these in a much simpler manner than has been done by Browning et al. We also show that it is essential to include the Coriolis term arising from the component of the earth's angular velocity acting tangential to the earths's surface in order to model the vertical velocity accurately.

A recent multigrid method for computing steady inviscid compressible flow is analyzed for the one-dimensional scalar case. The discretization in space by means of upwind differencing has first- or second-order accuracy. Three types of relaxation schemes that use only local information are examined. A numerical experiment with a smooth steady solution shows good agreement with the estimated damping rates. A discontinuous solution displays slower convergence. This is mainly caused by the singularity at the shock. The singularity can be removed by the additional constraint of local conservation, which is achieved through a special kind of local relaxation, using information about conservation from coarser grids. The sonic point causes some slow-down as well, but this can be neglected in practical applications. It turns out that a first-order-accurate solution can be obtained by 1 F-cycle per grid, whereas second-order accuracy requires about 4.

For problems defined on infinite domains, the most common computational methods use an artificial boundary gamma in order to make the computation finite. Since data are usually not available at this boundary, conditions must be used that somehow are related to the behavior of the solution outside the computational domain omega. This means that some apriori knowledge of the solution is necessary. In particular, if the system of differential equations has variable coefficients or is non-linear, then we shall make the coefficients constant outside omega. In this way there is a possibility to compute the solution exactly, or to compute it approximately in a simple way outside omega, such that boundary conditions can be derived on gamma. A previous work in the basic principles concerning artificial boundaries for hyperbolic systems were discussed. The one-dimensional case was analyzed thoroughly, and it was shown under what conditions it is possible to derive accurate conditions at the artificial boundary. A two-dimensional example was also treated, and a method leading to exact conditions for steady state solutions was presented. This paper at treats hyperbolic systems in two space-dimensions with non-zero initial data also outside omega.

In summary, graphs and s-expressions are really different manifestations of the same entities. While graphs are visual, they nonetheless possess dual representations in the textual s-expression alphabet. As a result, LISP, which is based entirely on s-expressions, is an ideal language for dealing with applications that perform extensive manipulations on graph-based objects. Scientific applications which make extensive use of both graph based objects as well as traditional numerical arrays can benefit by implementating their higher level data and control structures in LISP. Segments of code that are computationally intensive can be written in a static language such as C or FORTRAN and linked in with the higher level code. Such a design strategy could preserve the benefits of static compilation (i, e, . faster execution speed) while retaining the power and flexibility needed to manipulate complex data structures.