TRANSPORT_ODE_fct_evaluation

(experimental) switch for additional function evaluation within the implicit time integration scheme in EULERIMPL setting

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.
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TRANSPORT_ODE_fct_evaluation (experimental) switch for additional function evaluation within the implicit time integration scheme in EULERIMPL setting
EULERIMPL Higher order implicit Eulerian or ALE motion (recommended among the Euler implementations)