**Matrix: Fluorem/GT01R**

Description: GT01R: 2D inviscid case. F. Pacull, Lyon, France

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

Matrix properties | |

number of rows | 7,980 |

number of columns | 7,980 |

nonzeros | 430,909 |

structural full rank? | yes |

structural rank | 7,980 |

# of blocks from dmperm | 4 |

# strongly connected comp. | 4 |

explicit zero entries | 0 |

nonzero pattern symmetry | 88% |

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 7980-by-1 |

x | full 7980-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: GT01R: 2D inviscid case number of mesh nodes: 1596 block size: 5 variables: [rho,rho*u,rho*v,rho*w,rho*E] (rho w is "silent", the fourth row and column in each block can be removed) matrix order: 7980 nnz: 430909 comments: This is a 2D linear cascade turbine case. The grid corresponds to one inter-blade channel. The stencil involved by the convective scheme uses 9 nodes. Thus, there are 9 non-zero blocks for each node in the matrix. The specificity is that the computational domain is periodic, which introduces some non-zeros elements far away from the diagonal.

Ordering statistics: | result |

nnz(chol(P*(A+A'+s*I)*P')) with AMD | 1,446,641 |

Cholesky flop count | 3.3e+08 |

nnz(L+U), no partial pivoting, with AMD | 2,885,302 |

nnz(V) for QR, upper bound nnz(L) for LU, with COLAMD | 2,765,274 |

nnz(R) for QR, upper bound nnz(U) for LU, with COLAMD | 5,518,657 |

SVD-based statistics: | |

norm(A) | 195007 |

min(svd(A)) | 0.0308668 |

cond(A) | 6.31767e+06 |

rank(A) | 7,980 |

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.
*