TOL_v

(control of time step size) error tolerance for computing the velocity using SDIRK2 method in EULERIMPL setting

TOL_v = 1.0e-3
Default: TOL_v = 2.0e-3 The EULERIMPL scheme is a fully implicit method, which does not need to fulfill the CFL condition. Therefore the time step size is computed dependent on a given tolerance \begin{align} \textnormal{TOL} := \textnormal{RTOL} \, \| \mathbf{U}^k \| + \textnormal{ATOL}, \end{align} whereby in this case \( \textnormal{RTOL} = \textnormal{ATOL} =\) TOL_v is determined by the user. With the help of a proportional-integral (PI) controller the time step size is computed by \begin{align} \Delta t_{k+1} &= \begin{cases} \Delta t_{k} \left( \dfrac{\theta \, \textnormal{TOL}}{\| \mathbf{\hat{e}}_{k+1} \|} \right)^{\beta_{I}/(\hat{p}+1)} \left( \dfrac{\| \mathbf{\hat{e}}_{k} \|}{\| \mathbf{\hat{e}}_{k+1} \|} \right)^{\beta_{P}/(\hat{p}+1)}, & \dfrac{\| \mathbf{\hat{e}}_{k+1} \|}{\textnormal{TOL}} \leq 1.2 \\[18pt] \Delta t_{k} \left( \dfrac{\theta \, \textnormal{TOL}}{\| \mathbf{\hat{e}}_{k+1} \|} \right)^{1/(\hat{p}+1)}, & \dfrac{\| \mathbf{\hat{e}}_{k+1} \|}{\textnormal{TOL}} > 1.2\end{cases}. \end{align} The local error estimator \( \| \mathbf{\hat{e}}_{k+1} \| = \| \mathbf{U}^{k+1} - \mathbf{\hat{U}}^{k+1} \|\) is computed by using an embedded Runge-Kutta method where two results of different order are compared. Due to the use of the SDIRK2 method (2nd order) the result \( \mathbf{\hat{U}}^{k+1}\) is based on a method of order \( \hat{p} = 1\). The other parameters are \begin{align} \theta = 0.8, \qquad \beta_{I} = 0.3, \qquad \beta_{P} = 0.4. \end{align} Remark: For solving the ODE \( \mathbf{\dot{U}} = \mathbf{F}(t, \mathbf{U})\), the Singly Diagonally Implicit Runge Kutta (SDIRK2) 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} &= \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, \, \mathbf{U}^{k+1} \big) \Big), \end{align} \begin{align} \alpha = 1 - \frac{\sqrt{2}}{2} \end{align} is used, which is of second order accuracy. That is why it is abbreviated as SDIRK2. See time_integration_impl and time_integration_impl_solve_v. See TOL_T and TOL_keps.
This item is referenced in:
NB_OF_ACCEPTED_REPETITIONS number of permitted repetitions of substep in EULERIMPL setting
SUBSTEPS_IMPL number of implicit substeps with constant time step size in EULERIMPL setting
time_integration_impl order of implicit time integration scheme in EULERIMPL setting
time_integration_impl_solve_v order of implicit time integration scheme for velocity only (EULERIMPL)
TOL_keps (control of time step size) error tolerance for computing the k-epsilon model using SDIRK2 method in EULERIMPL setting
TOL_T (control of time step size) error tolerance for computing the temperature using SDIRK2 method in EULERIMPL setting
TOL_v (control of time step size) error tolerance for computing the velocity using SDIRK2 method in EULERIMPL setting
EULERIMPL Higher order implicit Eulerian or ALE motion (recommended among the Euler implementations)
TimeControl time control options