%-------------------------------------------------------------------------------
% UF Sparse Matrix Collection, Tim Davis
% http://www.cise.ufl.edu/research/sparse/matrices/Fluorem/DK01R
% name: Fluorem/DK01R
% [DK01R: 1D turbulent case. F. Pacull, Lyon, France]
% id: 2334
% date: 2010
% author: F. Pacull
% ed: T. Davis
% fields: name title A b x id date author ed kind notes
% kind: computational fluid dynamics problem
%-------------------------------------------------------------------------------
% notes:
% CFD matrices from Francois Pacull, FLUOREM, in Lyon, France        
%                                                                    
% We are dealing with CFD and more precisely steady flow             
% parametrization. The equations involved are the compressible       
% Navier-Stokes ones (RANS).  These matrices are real, square and    
% indefinite, they correspond to the Jacobian with respect the       
% conservative fluid variables of the discretized governing          
% equations (finite-volume discretization). Thus they have a         
% block structure (corresponding to the mesh nodes: the block        
% size is the number of variables per mesh node), they are not       
% symmetric (however, their blockwise structure has a high level     
% of symmetry) and they often show some kind of hyperbolic           
% behavior. They have not been scaled or reordered.                  
%                                                                    
% They are generated through automatic differentiation of the        
% flow solver around a steady state. A right hand-side is also       
% given for each matrix: this represents the derivative of the       
% equations with respect to a parameter (of operation or shape).     
% Since they are generated automatically, they may have "silent"     
% variables: these are variables corresponding to an identity        
% submatrix associated with a null right hand-side, for example      
% one of the three velocity components in a 2D case, or the          
% turbulent variables in a "frozen" turbulence case.                 
%                                                                    
% We believe that these matrices are good test cases when            
% studying preconditioning methods for iterative methods, such as    
% block incomplete factorization, or when studying domain            
% decomposition methods or deflation. They are actually being        
% studied by a few researchers in France regarding numerical         
% methods, through the LIBRAERO research project of the ANR (national
% research agency): ANR-07-TLOG-011.                                 
%                                                                    
% Francois Pacull, Lyon, France.  fpacull at fluorem.com             
%                                                                    
% Specific problem descriptions:                                     
%     DK01R: 1D turbulent case                                       
%     number of mesh nodes: 129                                      
%     block size: 7                                                  
%     variables: [rho,rho*u,rho*v,rho*w,rho*E,rho*k,rho*omega]       
%     (rho v and rho w are "silent", the third and fourth rows       
%     and columns                                                    
%     in each block can be removed)                                  
%     matrix order: 903                                              
%     nnz: 11758                                                     
%     comments: The DK01R matrix corresponds to a small 1D turbulent 
%     case. The grid has 129 nodes, non-uniformly spaced             
%     (geometrical distribution). The number of unknowns per node is 
%     7, leading to a linear system of 903 real algebraic equations. 
%     The 1D discretization of the partial differential equations    
%     uses a 5 points stencil, leading to a block penta-diagonal     
%     matrix, each block having size 7 by 7. Each diagonal block is  
%     related to two up- and two down-stream neighboring nodes,      
%     corresponding respectively to the 14 upper and 14 lower matrix 
%     rows, the node ordering being coherent with the 1D spatial     
%     node distribution. The stationary flow on which the matrix is  
%     based on is dominated by advection, characterized by a Mach    
%     number around 0.3.                                             
%-------------------------------------------------------------------------------
