DK01R: 1D turbulent case. F. Pacull, Lyon, France
Name DK01R
Group Fluorem
Matrix ID 2334
Num Rows 903
Num Cols 903
Nonzeros 11,766
Pattern Entries 11,766
Kind Computational Fluid Dynamics Problem
Symmetric No
Date 2010
Author F. Pacull
Editor T. Davis
Structural Rank 903
Structural Rank Full true
Num Dmperm Blocks 8
Strongly Connect Components 8
Num Explicit Zeros 0
Pattern Symmetry 96.1%
Numeric Symmetry 0%
Cholesky Candidate no
Positive Definite no
Type real
SVD Statistics
Matrix Norm 9.775521e+06
Minimum Singular Value 1.660209e-01
Condition Number 5.888126e+07
Rank 903
sprank(A)-rank(A) 0
Null Space Dimension 0
Full Numerical Rank? yes
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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.