Description: Atmospheric models, Andrei Bourchtein
|(undirected graph drawing)|
|number of rows||1,489,752|
|number of columns||1,489,752|
|structural full rank?||yes|
|# of blocks from dmperm||1|
|# strongly connected comp.||1|
|explicit zero entries||0|
|nonzero pattern symmetry||symmetric|
|numeric value symmetry||67%|
|kind||computational fluid dynamics problem|
|Additional fields||size and type|
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.
|nnz(chol(P*(A+A'+s*I)*P')) with AMD||1,937,851,538|
|Cholesky flop count||2.1e+13|
|nnz(L+U), no partial pivoting, with AMD||3,874,213,324|
|nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD||4,419,580,954|
|nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD||7,841,476,026|
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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.