Default:
TRANSPORT_ODE_fct_evaluation = 'NON'
If DIRK(Diagonally Implicit Runge-Kutta) methods of the form
\begin{align}
\boldsymbol{\eta}_j &= \mathbf{U}^{k} + \Delta t \sum \limits_{\nu=1}^{j} a_{j\nu} \, \mathbf{L}_{\nu}, \qquad j = 1, \ldots , s, \\ \mathbf{U}^{k+1} &= \mathbf{U}^{k} + \Delta t \sum \limits_{j=1}^{s} b_{j} \, \mathbf{L}_{j}
\end{align}
\begin{align}
\mathbf{L}_j = \mathbf{F}\left(t^k + c_j \, \Delta t, \, \boldsymbol{\eta}_j \right)
\end{align}
are used for solving the
ODE \( \mathbf{\dot{U}} = \mathbf{F}(t, \mathbf{U})\), each stage \( \boldsymbol{\eta}_j\) can be calculated one by one using the previous stages \( \boldsymbol{\eta}_{\nu}, \nu = 1,\ldots j-1\).
This requires the function values \( \mathbf{L}_j\) that are not yet directly available after the solution of the equation system for \( \boldsymbol{\eta}_j\).
Either one can evaluate the discretization function \( \mathbf{F}\) at \( \boldsymbol{\eta}_j\) (
TRANSPORT_ODE_fct_evaluation = 'YES'), what could be quite expensive, or after solving the
equation system one can use the relation (
TRANSPORT_ODE_fct_evaluation = 'NON')
\begin{align}
\mathbf{L}_j = \frac{1}{a_{jj} \Delta t} \left( \boldsymbol{\eta}_j - \mathbf{U}^{k} - \Delta t \sum \limits_{\nu=1}^{j-1} a_{j\nu} \, \mathbf{L}_{\nu} \right), \qquad j = 1, \ldots , s.
\end{align}
Due to the linearization of the discretization function both approaches are not equivalent. In some cases the additional evaluation
(
TRANSPORT_ODE_fct_evaluation = 'YES') can lead to more accurate results, but especially when using larger time steps it can become very unstable.
It has been observed that using
TRANSPORT_ODE_fct_evaluation = 'YES' brings no advantage in solving the velocity and k-epsilon model.
Therefore it is only implemented for the temperature. Thus
TRANSPORT_ODE_fct_evaluation = 'YES' only influences the temperature,
but for stability reasons it is recommended to use
TRANSPORT_ODE_fct_evaluation = 'NON'.
Remark: For solving the
ODE \( \mathbf{\dot{U}} = \mathbf{F}(t, \mathbf{U})\), the Singly Diagonally Implicit Runge-Kutta(SDIRK) method
\begin{align}
\boldsymbol{\eta}_1 &= \mathbf{U}^k + \Delta t \, \alpha \, \mathbf{F}\big(t^k + \alpha \, \Delta t, \, \boldsymbol{\eta}_1\big), \\[10pt] \mathbf{U}^{k+1} &= \boldsymbol{\eta}_2 = \mathbf{U}^k + \Delta t \, \Big((1-\alpha) \, \mathbf{F}\big(t^k + \alpha \, \Delta t, \, \boldsymbol{\eta}_1\big) + \alpha \, \mathbf{F}\big(t^k + \Delta t, \, \boldsymbol{\eta}_2 \big) \Big),
\end{align}
\begin{align}
\alpha = 1 - \frac{\sqrt{2}}{2}
\end{align}
is used, which is of second order accuracy. That's why it is abbreviated as SDIRK2.