5.1 Structure of matrices

As described in Sec. 3.4 , a matrix equation for solution variable eqn contains of a set of coefficients eqn where each row eqn corresponds to the linear equation for the cell with index eqn as follows:

2 3 2 3 2 3 a1;1 a1;2 a1;3 a1;N 1 b1 66 77 66 77 66 77 66 a2;1 a2;2 a2;3 a2;N 77 66 2 77 66 b2 77 66 a3;1 a3;2 a3;3 a3;N 77 66 3 77 = 66 b3 77 66 : : : : : 77 66 : 77 66 : 77 64 :: :: :: :: :: 75 64 :: 75 64 :: 75 aN;1 aN;2 aN;3 aN;N N bN \relax \special {t4ht=

Matrix sparsity

Imagine creating a matrix equation for a transport equation for a scalar field, e.g. Eq. (2.65 ) with zero heat source eqn.

PICT\relax \special {t4ht=

The figure shows a matrix of size eqn, where eqn is the number of cells. Circles denote non-zero coefficients, which are filled for the diagonal coefficients (eqn). The matrix eqn is sparse, i.e. the majority of the coefficients eqn are 0 (zero).

The sparsity is due to each cell interacting only with adjacent cells connected through its faces. For example, a 3D mesh of hexahedral cells produces up to 7 coefficients per matrix row, with one diagonal coefficient corresponding to a particular cell and 6 off-diagonal coefficients for the neighbour cells.

Matrix (a)symmetry

A symmetric matrix possesses the same coefficients across the diagonal, i.e. eqn. The discretisation of a Laplacian term, e.g. eqn, described in Sec. 3.7 , produces coefficients that are symmetric because the surface normal gradient Eq. (3.5 ) uses the current and neighbour cell values in equal measure.

PICT\relax \special {t4ht=

However, the discretisation of an advection term, e.g. eqn, generally produces asymmetric coefficients. For example, if the upwind scheme is applied for flow from cell 1 into cell 2, then there would a contribution to eqn but not to eqn.

Matrix size

Parallel simulation allows affordable solution on huge meshes. For a mesh of size eqn, the matrix would have eqn coefficients, typically with eqn that are are non-zero.

For reasons of efficiency, zero coefficients are not stored in the computer’s memory. Instead the storage provides an array of non-zero coefficients and addressing arrays of the corresponding row and column indices for each coefficient.

Notes on CFD: General Principles - 5.1 Structure of matrices