Fractional step scheme

Beside the three velocity components, the discretized momentum balance equations (2.41) for a three dimensional case includes pressure as a coupling variable. A possible approach to decouple them is the application of so-called fractional step method. The main idea of this method consists in introducing an intermediate velocity as supplementary variable and splitting the momentum equation. The modification introduced by R.Codina [#!Codina01!#] splits the the discretized time step is split into two sub-steps. The implicit part of the pressure is avoided and assigned to the second step.

$$\mbox{$\displaystyle\frac{\partial \,{u_i}}{\partial \,{t}}$} \approx \frac{u^{n+1}_i-u^n_i}{\Delta t}=\frac{u^{n+1}_i-u^*_i+u^... ...ial x_j}\left(\frac{\partial u_i}{\partial x_j}\right)+b_i\right]^{n+\theta}\;.$$ (2.42)

where the intermediate velocity $u^*_i$ is introduced. Splitting the equation in the following manner gives the expression for the unknown velocities

$$u^*_i = u^n_i+b_i\Delta t - \frac{\Delta t}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} \gamma p^n+\frac{\Delta t\mu}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_j}}$} \left(\mbox{$\displaystyle\frac{\partial \,{u^{n+\theta}_i}}{\partial \,{x_j}}$}\right)\;,$$ (2.43)

$$u^{n+1}_i = u^*_i- \frac{\Delta t}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} (p^{n+1}-\gamma p^n)\;.$$ (2.44)

The pressure split is here introduced by the new parameter $\gamma$ defining the amount of splitting and can take values from 0 to 1. The body loads are considered to be constant over time step.

In a similar way, the fractional step method is applied on the mass conservation equation. Here, the time derivative of density would be approximated. As we examine an incompressible flow, whose density does not change in time, merely the intermediate velocity term is incorporated in the divergence of the velocity.

$$\mbox{$\displaystyle\frac{\partial \,{(u^{n+1}_i-u^*_i+u^*_i)}}{\partial \,{x_i}}$} = 0 \;,$$ (2.45)

which can be decomposed into two sub-equations

$$\mbox{$\displaystyle\frac{\partial \,{u^*_i}}{\partial \,{x_i}}$} =$$ 0 (2.46)

$$\mbox{$\displaystyle\frac{\partial \,{(u^{n+1}_i-u^*_i)}}{\partial \,{x_i}}$} = 0\;.$$ (2.47)

By substituting for the velocity difference into the equation (2.44) we obtain

$$\mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} (u^{n+1}_i - u^*_i) = \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} \left(-\frac{\Delta t}{\rho}\mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$}(p^{n+1}-\gamma p^n)\right)\;.$$ (2.48)

Now we can sum the separated mass equations together. This operation gives the coupled mass-momentum equation

$$\mbox{$\displaystyle\frac{\partial \,{u^*_i}}{\partial \,{x_i}}$} - \frac{\Delta t}{\rho}\frac{\partial^2}{\partial x^2_i}(p^{n+1}-\gamma p^n) = 0\;.$$ (2.49)

The final set of equations reads

$$u^*_i = u^n_i+b_i\Delta t - \frac{\Delta t}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} \gamma p^n+\frac{\Delta t\mu}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_j}}$} \left(\mbox{$\displaystyle\frac{\partial \,{u^{n+\theta}_i}}{\partial \,{x_j}}$}\right)\;,$$ (2.50)

$$\frac{\partial^2}{\partial x^2_i}(p^{n+1}) = \frac{\rho}{\Delta t} \mbox{$\displaystyle\frac{\partial \,{u^*_i}}{\partial \,{x_i}}$} +\frac{\partial^2}{\partial x^2_i}(\gamma p^n)\;,$$ (2.51)

$$u^{n+1}_i = u^*_i- \frac{\Delta t}{\rho} \mbox{$\displaystyle\frac{\partial \,{}}{\partial \,{x_i}}$} (p^{n+1}-\gamma p^n)\;.$$ (2.52)

The above PFEM formulation is based on the paper by Idelsohn, Oñate and Del Pin [#!Idelsohn04!#]. The authors described an approach using arbitrary time discretization scheme and pressure split factor. Their choice of implicit scheme $\theta=1$ was motivated by better convergence properties, whereas the decision for $\gamma = 0$ leading to greater pressure split was driven by better pressure stabilization.