time_integration_impl

order of implicit time integration scheme in EULERIMPL setting

Default: time_integration_impl = 2
  • time_integration_impl = 1 uses for time integration the implicit Euler method, which is of first order accuracy.
  • time_integration_impl = 2 uses for time integration the implicit SDIRK2 (Singly Diagonally Implicit Runge-Kutta) method, which is of second order accuracy.
SDIRK2 method is only used if at least one phase (see KindOfProblem) is calculated with EULERIMPL. Using LAGRANGE it is not worth to use second order time integration because it does not improve the accuracy of the results. Due to the movement of the points in the LAGRANGE setting only viscous parts are implicitly solved. Thus the time discretization error plays a minor part. Therefore time_integration_impl is automatically set to 1 in LAGRANGE phase. Remark: For solving the ODE \( \mathbf{\dot{U}} = \mathbf{F}(t, \mathbf{U})\), the SDIRK2 method is \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} The time step size is controlled by the error tolerances TOL_T (temperature), TOL_v (velocity) and TOL_keps (k-epsilon model).
This item is referenced in:
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)
ImplicitEuler ImplicitEuler
SDIRK2 SDIRK2