**Matrix: Fluorem/DK01R**

Description: DK01R: 1D turbulent case. F. Pacull, Lyon, France

(bipartite graph drawing) | (graph drawing of A+A') |

Matrix properties | |

number of rows | 903 |

number of columns | 903 |

nonzeros | 11,766 |

structural full rank? | yes |

structural rank | 903 |

# of blocks from dmperm | 8 |

# strongly connected comp. | 8 |

explicit zero entries | 0 |

nonzero pattern symmetry | 96% |

numeric value symmetry | 0% |

type | real |

structure | unsymmetric |

Cholesky candidate? | no |

positive definite? | no |

author | F. Pacull |

editor | T. Davis |

date | 2010 |

kind | computational fluid dynamics problem |

2D/3D problem? | yes |

Additional fields | size and type |

b | full 903-by-1 |

x | full 903-by-1 |

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.

Ordering statistics: | result |

nnz(chol(P*(A+A'+s*I)*P')) with AMD | 7,807 |

Cholesky flop count | 8.0e+04 |

nnz(L+U), no partial pivoting, with AMD | 14,711 |

nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD | 7,878 |

nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD | 14,608 |

SVD-based statistics: | |

norm(A) | 9.77552e+06 |

min(svd(A)) | 0.166021 |

cond(A) | 5.88813e+07 |

rank(A) | 903 |

sprank(A)-rank(A) | 0 |

null space dimension | 0 |

full numerical rank? | yes |

singular values (MAT file): | click here |

SVD method used: | s = svd (full (A)) ; |

status: | ok |

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.
*