Global residuals

TRACE computes different residuals for the different solver modes.

Non-linear time domain solver

For the non-linear time domain solver, the residual is given in three different norms as indicated in the suffix of the Residual output variable. The reduction to a global scalar value starts from the residual vector \(R_i^j\) where \(i\) is one of the 5 governing equations for each cell \(j\) in the domain. The residual is given by

flux balances - volume integral of sources + temporal terms (in case of unsteady simulations)

The cell volume is denoted by \(V_j\), the local pseudo-time step size by \(\Delta \tau_j\) and the CFL number by \(c\).

ResidualL1

\begin{equation} \frac{1}{n_\mathrm{cells}} \sum_j^{n_\mathrm{cells}} \sum_i^{n_\mathrm{equations}} \left| \frac{\Delta \tau_j}{V_j c} R_i^j \right| \end{equation}

ResidualL2

\begin{equation} \sqrt{\frac{1}{n_\mathrm{cells}} \sum_j^{n_\mathrm{cells}} \sum_i^{n_\mathrm{equations}} \left( \frac{\Delta \tau_j}{V_j c} R_i^j \right)^2} \end{equation}

ResidualMax

\begin{equation} \max_j \sum_i^{n_\mathrm{equations}} \left| \frac{\Delta \tau_j}{V_j c} R_i^j \right| \end{equation}

Here also the block and cell indices (BlockIndex, CellIndexI, CellIndexJ, CellIndexK) where the maximal residual was detected are written.

For the other transported variables, such as turbulence kinetic energy or turbulent dissipation rate, the global residual in the L1 and Max norm is computed in the same way for each equation separately ( \({n_\mathrm{equations}} = 1\)).

Note

Note that this is all based on non-dimensional quantities. See Non-dimensionalization.