# A Semi-Explicit Multi-Step Method for Solving Incompressible Navier-Stokes Equations
## 2 Governing Equations
### Fractional Step Split  
The governing system consists of momentum Equation (53) and the mass conservation equation (Equation (37)), which in the time-integrated form is written as:  
\[\bar{\mathbf{D}}\bar{\mathbf{u}}_{n+1}+\mathbf{L}_{\tau}\bar{\mathbf{p}}_{n+ 1}=\hat{\mathbf{F}}. \tag{59}\]  
For decoupling the velocity from the pressure, we shall apply the fractional step split by introducing auxiliary velocity \(\bar{\mathbf{u}}\), which is non-solenoidal.  
The fractional split applied to Equation (53) leads to:  
\[\mathbf{M}\frac{\bar{\mathbf{u}}-\bar{\mathbf{u}}^{n}}{\Delta t}= \bar{\mathbf{R}}+\gamma\bar{\mathbf{G}}\bar{\mathbf{p}}_{n}, \tag{60}\] \[\mathbf{M}\frac{\bar{\mathbf{u}}_{n+1}-\bar{\mathbf{u}}}{\Delta t }=\frac{1}{2}\bar{\mathbf{G}}\bar{\mathbf{p}}_{n+1}+\frac{1}{2}\bar{\mathbf{G }}\bar{\mathbf{p}}_{n}-\gamma\bar{\mathbf{G}}\bar{\mathbf{p}}_{n}. \tag{61}\]  
where \(\gamma=0\) (no pressure gradient in the fractional momentum equations) or \(\gamma=1\) (historical pressure gradient in the fractional momentum equation). The residuals are defined as:  
\[\bar{\mathbf{R}}=\frac{1}{6}[\bar{\mathbf{r}}_{1}+2\bar{\mathbf{r} }_{2}+2\bar{\mathbf{r}}_{3}+\bar{\mathbf{r}}_{4}],\] (62) \[\bar{\mathbf{r}}_{1}=\bar{\mathbf{F}}_{n}-\bar{\mathbf{C}}(\bar{ \mathbf{u}}_{n})(\bar{\mathbf{u}}_{n}),\] (63) \[\bar{\mathbf{r}}_{2}=\bar{\mathbf{F}}_{\alpha  
convective operator can be computed using the fractional velocity \(\tilde{\mathbf{C}}(\tilde{\mathbf{u}}_{n+1})\approx\tilde{\mathbf{C}}(\tilde{ \mathbf{u}})\). This means that convection is computed using the auxiliary non-solenoidal velocity. Following the ideas of [10], the resulting accuracy can be estimated. The corresponding splitting error depends on \(\gamma\): \(O(||\tilde{\mathbf{u}}_{n+1}-\tilde{\mathbf{u}}||)=\Delta tO(||0.5(p_{n+1}+p_ {n})-\gamma p_{n}||)\). If \(\gamma=0\), the order of approximation is \(\Delta t\). If \(\gamma=1\), the order is \(\Delta t^{2}\) since \(O(||p_{n+1}-\gamma p_{n}||)=\Delta t\). For higher accuracy, a second option is used here, thus choosing \(\gamma=1\).  
Applying the mass conservation Equation (59) to Equation (61) leads to:  
\[\Delta t\mathbf{DM}^{-1}\left(\frac{1}{2}\mathbf{G}\tilde{\mathbf{p}}_{n+1}- \frac{1}{2}\mathbf{G}\tilde{\mathbf{p}}_{n}\right)=\tilde{\mathbf{D}}\tilde{ \mathbf{u}}-\tilde{\mathbf{F}}-\mathbf{L}_{\tau}\tilde{\mathbf{p}}_{n+1}. \tag{67}\]  
Introducing the commonly-used approximation of the Laplace operator \(\mathbf{DM}^{-1}\mathbf{G}\approx\mathbf{L}\), one obtains:  
\[\mathbf{M}\frac{\tilde{\mathbf{u}}-\tilde{\mathbf{u}}^{n}}{\Delta t}=\tilde{ \mathbf{R}}+\tilde{\mathbf{G}}\tilde{\mathbf{p}}_{n}, \tag{68}\]  
\[\frac{\Delta t}{2}\mathbf{L}(\tilde{\mathbf{p}}_{n+1}-\tilde{\mathbf{p}}_{n}) =\tilde{\mathbf{D}}\tilde{\mathbf{u}}-\tilde{\mathbf{F}}-\mathbf{L}_{\tau} \tilde{\mathbf{p}}_{n+1}, \tag{69}\]  
\[\mathbf{M}\frac{\tilde{\mathbf{u}}^{n+1}-\tilde{\mathbf{u}}}{\Delta t}-\frac{ 1}{2}\tilde{\mathbf{G}}(\tilde{\mathbf{p}}_{n+1}-\tilde{\mathbf{p}}_{n})=0. \tag{70}\]  
which defines our discrete model. Equation (69) is known as the \"pressure Poisson equation\". The problem is thus solved in three steps:  
1. Fractional momentum (Equation (68)) is solved. Fractional velocity \(\tilde{\mathbf{u}}\) is obtained (explicit step).
2. The equation for the pressure (Equation (69)) is solved. Pressure \(\tilde{\mathbf{p}}_{n+1}\) is obtained (implicit step).
3. Velocity is corrected (Equation (70)) giving \(\tilde{\mathbf{u}}_{n+1}\) (explicit step).  
The accuracy and efficiency of the method proposed here will be estimated by means of a numerical test.