 The open source CFD toolbox
Residuals

# Tolerances

Equation tolerances are described in terms of absolute and relative quantities:

tolerance       1e-6;
relTol          0.1;


If the equation initial residual satisfies either of the specified values, the system of equations are assumed solved and will not be evolved.

# Calculation

The residual calculation is solver-specific. However, the general approach follows:

For a matrix system

$\mat{A} \vec{x} = \vec{b},$

the residual is defined as

$\vec{r} = \vec{b} - \mat{A} \vec{x}.$

We then apply residual scaling using the following normalisation procedure:

$n = \sum \left( \mag{\mat{A}\vec{x} - \mat{A}\av{\vec{x}}} + \mag{\vec{b} - \mat{A}\av{\vec{x}}} \right)$

where $$\av{\vec{x}}$$ is the average of the solution vector. The scaled residual is finally given by:

$r = \frac{1}{n} \sum \mag{\vec{b} - \mat{A} \vec{x}}.$

This form leads to a normalised residual of 1 for uniform systems, i.e. where $$\vec{x} = \av{\vec{x}}$$. However, this also shows that if the initial solution changes, e.g. using non-uniform conditions, the normalisation also changes, leading to a different convergence history.

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