I'm doing CFD simulations for blood flow in unstructured grids. My boundary condition at the outlets is called three-element Windkessel which basically calculates the pressure by solving this ODE:
$$R_{2}C\frac{dP}{dt} + P(t) = C R_{1}R_{2}\frac{dQ}{dt} + (R_{1}+R_{2})Q(t)$$
Where $P(t)$ is the pressure at the outlet, $Q(t)$ is the flux at the outlet, and $R_{1}$, $R_{2}$, and $C$ are known constants. My initial condition is: $P(0) = 0$. I discretized this ODE as:
$$P^{t+\Delta t} = \frac{1}{2\Delta t + R_{2}C}(R_{2}CP^{t-\Delta t}+2CR_{1}R_{2}(Q^{t}-Q^{t-\Delta t})+2(R_{1}+R_{2})\Delta tQ^{t})$$
My problem is that because my initial condition is set to $P(0) = 0$, it takes very long time to reach cyclic stability (my flux is periodic due to heart beat), which is very expensive computationally or even impossible for me to reach. Is there any way of having a better guess for initial condition here in order to make sure I can reach the cyclic stability faster?