Simulate with complex geometries and complex physics
SDIRK2
This algorithm solves the semi-discrete ODE-system
\begin{align} \mathbf{\dot{U}}(t) = \mathbf{F}(t,\mathbf{U}(t))\end{align}
with the implicit SDIRK2 method which is of order 2:
\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 ODE-system comes from the spatial discretization of velocity and pressure in solve_V_2.
Both vp- and v-- can be solved.
The time integration scheme can be controlled by time_integration_impl and resp. for the velocity by time_integration_impl_solve_v.