%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/Bourchtein/atmosmodm
% name: Bourchtein/atmosmodm
% [Atmospheric models, Andrei Bourchtein]
% id: 2268
% date: 2009
% author: A. Bourchtein
% ed: T. Davis
% fields: name title A b id date author ed kind notes
% kind: computational fluid dynamics problem
%-------------------------------------------------------------------------------
% notes:
% 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.                                             
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