Matrix: Bourchtein/atmosmodl

Description: Atmospheric models, Andrei Bourchtein

Bourchtein/atmosmodl graph
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  • download as a MATLAB mat-file, file size: 34 MB. Use UFget(2267) or UFget('Bourchtein/atmosmodl') in MATLAB.
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  • download in Rutherford/Boeing format, file size: 40 MB.

    Matrix properties
    number of rows1,489,752
    number of columns1,489,752
    structural full rank?yes
    structural rank1,489,752
    # of blocks from dmperm1
    # strongly connected comp.1
    explicit zero entries0
    nonzero pattern symmetrysymmetric
    numeric value symmetry 67%
    Cholesky candidate?no
    positive definite?no

    authorA. Bourchtein
    editorT. Davis
    kindcomputational fluid dynamics problem
    2D/3D problem?yes

    Additional fieldssize and type
    bfull 1489752-by-2


    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.                                             

    Ordering statistics:result
    nnz(chol(P*(A+A'+s*I)*P')) with AMD1,937,851,538
    Cholesky flop count2.1e+13
    nnz(L+U), no partial pivoting, with AMD3,874,213,324
    nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD4,419,580,954
    nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD7,841,476,026

    For a description of the statistics displayed above, click here.

    Maintained by Tim Davis, last updated 12-Mar-2014.
    Matrix pictures by cspy, a MATLAB function in the CSparse package.
    Matrix graphs by Yifan Hu, AT&T Labs Visualization Group.