## 5.9 Systems of equations

Most CFD calculations involve solving a system of equations that represent the physics of the problem. For example, laminar ﬂow by natural convection can be represented by the equations introduced in Sec. 2.20 , reproduced below.

The system provides 3 equations (1 vector, 2 scalar) which can be solved for 3 unknowns, , and .

As discussed in Sec. 3.4 , the ﬁnite volume method solves an individual matrix equation for each variable, e.g. for . The vector equation for is decoupled into 3 matrix equations for individual components, i.e. , and .

Each individual matrix equation for one solution variable, e.g. , incorporates terms from other variables, e.g. , into the source vector . The contribution to is calculated using current values of the respective variables. Systems of equations are thereby solved by successive substitution of solved variables into the source vectors of subsequent equations.

An iterative solution for a single equation, like the one in Sec. 5.7 , can be extended to a system of equations. Time is incremented by and equations are solved in sequence, before returning to start the next time step with the increment of .

The substitutions in the momentum and pressure equations are particularly important, culminating in corrections to and the advective ﬂux , discussed in Sec. 5.10 .

At the start of any time step the current becomes for the discretisation of the term in the momentum equation. The advection term is discretised by Eq. (3.8 ), treating one as ﬂux and the other as the advected quantity.

The equation is solved for . The new solution for is substituted into the equation which is solved for . The new solution for is then used to correct in order to help enforce the mass conservation constraint ().

Before the current solution step is completed, is also corrected to reduce the error in the discretisation of when it then becomes in the following solution step. The correction also provides a better “initial guess” for the matrix solution of the next momentum equation, which reduces the solution time.