Atmospheric modeling problems from Andrei Bourchtein

These matrices arise in the numerical weather prediction and atmospheric
modeling.  Such matrices usually appear in semi-implicit schemes applied to
three-dimensional Euler or Navier-Stokes equations (called nonhydrostatic
models in the atmospheric sciences) or to their simplified form with
hydrostatic balance equation instead of vertical momentum equation (called
hydrostatic or primitive equation models in the atmospheric sciences).

Such linear systems represent discretization of three-dimensional elliptic
problems (frequently Dirichlet or Neumann problems for Helmholtz or
quasi-Helmholtz equations), arising at each time step of semi-implicit
algorithms due to implicit time approximation of some linear terms in the
governing equations.  If spectral spatial approximation is applied, then the
elliptic problem is usually transformed to the linear system with a diagonal
matrix solved trivially.  If finite-difference or finite-element approximation
is used, then the linear systems with the sparse matrices of the coefficients
similar to the four submitted matrices arise.  However, the semi-implicit
schemes usually do not require explicit construction of the matrix of
coefficients, neither do iterative methods used to solve these systems in the
atmospheric models.  Besides, avoiding construction of the matrix of
coefficients allows reducing the required computer memory.  Due to these
reasons, as far as I know, the explicit form of matrices of coefficients
usually is not described, except for the local structure of the respective
difference equations.

The two right-hand sides b(:,1) and b(:,2) refer to the long wave or
short wave perturbation of atmospheric fields, respectively.

The description of such semi-implicit algorithms together with arising elliptic
problems can be found, for example, in the following recent papers (and
references therein):

1. Steppeler J., Hess R., Schattler U., Bonaventura L.: Review of numerical
methods for nonhydrostatic weather prediction models. Met. Atm Phys. 82 (2003)
287-301.

    This is a review paper on nonhydrostatic models, including particularly,
    semi-implicit time differencing.  Some description of arising elliptic
    problems and their solvers used in atmospheric models can be found on
    pp.294-296.

2. Cote J., Gravel S., Methot A., Patoine A., Roch M., Staniforth A.: The CMC-MRB
global environmental multiscale (GEM) model. Part I: Design considerations and
formulation. Mon. Wea. Rev. 126 (1998) 1373-1395.

3. Yeh K.S., Cote J., Gravel S., Methot A., Patoine A., Roch M., Staniforth A.: The
CMC-MRB global environmental multiscale (GEM) model. Part III: Nonhydrostatic
formulation. Mon. Wea. Rev. 130 (2002) 339-356.

    This pair of papers is about hydrostatic and nonhydrostatic versions of the
    modern semi-implicit Canadian model.  Some description of elliptic problems
    and their solution can be found on p.1389 of the first paper and p.343 of
    the second paper.

4. Davies T., Cullen M.J.P., Malcolm A.J., Mawson M.H., Staniforths A., White A.A.,
Wood N.: A new dynamical core for the Met Office’s global and regional modeling.
Q. J. Roy. Met. Soc. 131 (2005) 1759-1782.

    This is a brief report on United Kingdom modern semi-implicit model.  Some
    description of elliptic problem can be found on p.1778 and its solution
    on pp.1771-1772.