3.4 Equation discretisation

Equation discretisation converts partial differential equations for continuous fields, e.g. pressure eqn, into sets of linear equations for discrete fields.

The values of the principal fields, e.g. eqn, are associated with cells. A field is then represented by an array of values, eqn, for cell indices eqn. eqn is the total number of cells.

PICT\relax \special {t4ht=

Equation discretisation creates a linear equation for each cell. For cell 43 above, the equation might have the following form:

pict\relax \special {t4ht=

where eqn and eqn are coefficients corresponding to cell indices eqn,eqn (diagonal coefficient in bold). The set of linear equations for all cells can be written as a matrix equation of the form:

2 32 3 2 3 a1;1 a1;2 a1;3 a1;N p1 b1 66 7766 77 66 77 66 a2;1 a2;2 a2;3 a2;N 7766 p2 77 66 b2 77 66 a3;1 a3;2 a3;3 a3;N 7766 p3 77= 66 b3 77 66 : : : : : 7766 : 77 66 : 77 64 :: :: :: :: :: 7564 :: 75 64 :: 75 aN;1 aN;2 aN;3 aN;N pN bN \relax \special {t4ht=
The matrix contains of a set of coefficients eqn where each row eqn corresponds to the linear equation for the cell with index eqn. Each row of coefficients are non-zero only for the respective cell (diagonal eqn) and near-neighbours. All other coefficients are zero, making the matrix extremely sparse.

The matrix equation can be represented as eqn, where: eqn are the matrix coefficients; eqn are the source coefficients eqn; and, eqn is the discretised pressure field. It may also be illustrated as follows, where ‘eqn’ indicates non-zero coefficients.

 ? ? ? ? ? ? ? p = ? ?? ?? ?? ?? \relax \special {t4ht=

Segregated solution

A CFD simulation generally solves a set of physical equations, e.g. for mass, momentum and energy conservation. The finite volume method traditionally discretises each physical equation separately to form individual matrix equations for single solution variables, e.g. eqn, eqn, eqn, rather than creating a single matrix equation that represents all the physical equations.

The segregated matrix equations are solved one variable at a time, e.g. solving for eqn, eqn and eqn in separate steps. Where the solution variable is a vector or tensor, e.g. eqn, it is decoupled into individual matrix equations for each component, e.g. eqn, eqn, eqn.

The matrix equations are solved in an iterative sequence, in which the equation for one variable, e.g. eqn, incorporates current values of other variables, e.g. eqn, eqn and eqn, into the source vector, as shown below.

PICT\relax \special {t4ht=

Notes on CFD: General Principles - 3.4 Equation discretisation