Numerical models of the atmosphere use meshes to describe fields
like density to temperature. In the most simple
case the mesh is one dimensional and has grid points x_i. A density
field h(x) is represented in the simplest case by grid point
values `h_i = h(x_i)`.
A very simple test case is the the transport
of a density field in a homogeneous velocity field, where for the
analytic solution the density field is transported without
deformation. Numerical approximation procedures try to
approximate this process of transport of the density
field and due to approximation errors this transport is no longer
deformation free. Such approximation errors tend to be more accurate
when a structure is supported by many grid points x_i. The most
simple grid is that of constant mesh length `x_i+1 - x_i`.
Examples of
such procedures are described in our book (see [1]). For this simple uniform
grid the initial value and the numerically transported fields for
different times are shown in fig 2 . The numerical method is
described in detail in [1].

It is seen in fig 2 that the transport is rather accurate, though
deformations and a loss of amplitude produce errors in the advection process.
Such relatie accuracy of the transport process with many numerical
processes is obtained with regular meshes x_i where `x_i+1 - x_i` is
independent of i. Details are described in the MoW book (see [1]).
This book describes some of the numerical methods in detail and
describes also methods which maintain their accuracy when the grid
becomes irregular. Such accuracy with irregular grids is
dependent on parameters w_i depending on i. The software to
compute the w_i and programs to accurately compute deriatives
and the transport process for arbitrarily irregular grids are
given in the application part of this Web page.

Work in this application area
requires some basic knowledge of the programming language Fortran
and the operating system Unix/Linux. A **login**
to an account on our server will be
necessary.
On The Unix/Linux side (after login) a program similar to the one from
[1] will be given.

While the program in [1] requires a regular
one dimensional grid the Unix/Linux login offers a program for
**arbitrarily irregular grids**. Acurate results can be
obtained for irregular grids. Investigating the
results for different resolutions of
the same solution can be done and we expect a convergence of at
least third order.

References:

[1]

Contact: send mail to Juergen Steppeler