## MVMMCD: Model 2-D Convection Diffusion Operator

### from the NEP Collection

Matrix Generator MVMMCD
Source: unclear
Discipline: Computational fluid dynamics
Language: Fortran
Output format: matrix-vector multiply

This test matrix is from the following constant-coefficient convection diffusion equation which is widely used for testing and analyzing numerical methods for the solution of linear system of equations,

where p1, p2 and p3 are positive constants. Discretization by the finite difference scheme with a 5-point stencil on a uniform m × m grid gives rise to a sparse linear system of equations

where A is of order m2 and u and b are now vectors of size m2. Centered differences are used for the first derivatives. If the grid points are numbered using the row-wise natural ordering, then A is a block tridiagonal matrix of the form

,

with

,

where .

The spectral decomposition of the convection diffusion matrix is known. For , if is a diagonal matrix with ((i-1)m+j)th diagonal entry

,

then is symmetric. For , let (k,l) denote (k-1)m-l. Then the (k,l)th unnormalized eigenvector of has (i,j)th entry

.

The reader may now verify that the (k,l)th eigenvalue of A is

.

Figure 1 shows the eigenvalue distribution of 961 by 961 convection diffusion matrix with p1 = 25, p2 = 50 and p3 = 250.

Since the normalized eigenvectors of a symmetric matrix are orthogonal, the eigenvectors of A are graded like the diagonal of and the condition number of the eigenvalue problem for A is

Fortran calling sequence.

Parameters:

 N order of the matrix (must be the square of an integer) p1 coefficient of 2ux in differential operator p2 coefficient of 2uy in differential operator p3 coefficient of -u in differential operator

## Matrix Instances:

• CDDE1 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=1, p2=2, p3=30
• CDDE2 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=25, p2=50, p3=30
• CDDE3 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=1, p2=2, p3=80
• CDDE4 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=25, p2=50, p3=80
• CDDE5 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=1, p2=2, p3=250
• CDDE6 (real unsymmetric, 961 by 961, 4681 nonzeros) Parameters: N=961, p1=25, p2=50, p3=250

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