# A Semi-Explicit Multi-Step Method for Solving Incompressible Navier-Stokes Equations
## 2 Governing Equations
### Time Integration  
Taking into account the implicit nature of the pressure in the incompressible flows, a fully-explicit time integration scheme cannot be used (strictly speaking, a fully-explicit scheme can be defined, but the associated critical time step would be prohibitively small, being governed by the acoustic pressure scale). However, one attractive option is defined by integrating the momentum equation explicitly, except for the pressure term, which is treated in an implicit way. Explicit time integration techniques are generally faster than their implicit counterparts as they do not involve non-linear iterations, nor require solving linear systems (in case mass matrix lumping is used). However, they are conditionally stable and are restricted by a critical time step size, which may be prohibitively small. A powerful commonly-used class of explicit methods is the family of Runge-Kutta time integration schemes. These integration methods involve the evaluation of residuals of the governing equation at several sub-steps. Runge-Kutta schemes with \(n\) intermediate steps are characterized by the \(n\)-th order of accuracy for \(n\leq 4\). For larger values of sub-steps, the accuracy order is less than \(n\). This makes the four-step scheme the most commonly-implemented version. Moreover, considering the restrictions of the critical time step size, this version of the method offers a good balance between the sub-step number and the maximum permissible time step size. The details regarding the stability domain of Runge-Kutta schemes can be consulted in, e.g., [28] (Chapter 5.3). Considering that the solution of a given problem is known at \(t_{n}\), the Runge-Kutta method requires successive computation of the governing equation residuals at several sub-steps in order to obtain the solution at \(t_{n+1}\). For a general Cauchy problem of type:  
\[\frac{\partial y}{\partial t}=f(y(t),t), \tag{38}\]  
the Runge-Kutta scheme of fourth order provides:  
\[\frac{y_{n+1}-y_{n}}{\Delta t}=\frac{1}{6}[k_{1}+2k_{2}+2k_{3}+k_{4}], \tag{39}\]  
where \(k_{1},\ \ldots,\ k_{4}\) are the intermediate residuals or \"right-hand side\" corrections that are obtained as:  
\[k_{1}=f(t_{n},y_{n}) \tag{40}\] \[k_{2}=f(t_{n}+\frac{\Delta t}{2},y_{n}+\frac{\Delta t}{2}k_{1}),\] (41) \[k_{3}=f(t_{n}+\frac{\Delta t}{2},y_{n}+\frac{\Delta t}{2}k_{2}),\] (42) \[k_{4}=f(t_{n}+\Delta t,y_{n}+\Delta tk_{3}). \tag{43}\]  
Clearly, to compute \(y_{n+1}\), the function must be evaluated four times, at each intermediate step. It also involves four updates of the primary variable. Next, the momentum equation is integrated using this scheme. The corresponding residual can be defined as:  
\[\mathbf{r}=\bar{\mathbf{F}}-\bar{\mathbf{C}}\bar{\mathbf{u}}+\bar{\mathbf{G}} \bar{\mathbf{p}}. \tag{44}\]  
Thus, the set of velocities and residuals to be computed in order to integrate the momentum equation in time using Equations (40)-(43) reads (note that the pressure terms are treated implicitly, i.e., they correspond to the end-of-sub-step values):  
\[\mathbf{r}_{1}=\bar{\mathbf{F}}_{n}-\bar{\mathbf{C}}(\bar{\mathbf{u}}_{n})( \bar{\mathbf{u}}_{n})+\bar{\mathbf{G}}\bar{\mathbf{p}}_{\alpha 1}, \tag{45}\]  
\[\bar{\mathbf{u}}_{\alpha 1}=\bar{\mathbf{u}}_{n}+\frac{1}{2}\mathbf{M}^{-1} \Delta t\mathbf{r}_{1}, \tag{46}\]  
\[\mathbf{r}_{2}=\bar{\mathbf{F}}_{\alpha 1}-\bar{\mathbf{C}}(\bar{\mathbf{u}}_{ \alpha 1})(\bar{\mathbf{u}}_{\alpha 1})+\bar{\mathbf{G}}\bar{\mathbf{p}}_{\alpha 2}, \tag{47}\]  
\[\frac{y_{n+1}-y_{n}}{\Delta t}=\frac{1}{6}[k_{1}+2k_{2}+2k_{3}+k_{4}], \tag{39}\]0  
\[\frac{y_{n+1}-y_{n}}{\Delta t}=\frac{1}{6}[k_{1}+2k_{2}+2k_{3}+k_{4}], \tag{39}\]1  
\[\frac{y_{n+1}-y_{n}}{\Delta t}=\frac{1}{6}[k_{1}+2k_{2}+2k_{3}+k_{4}], \tag{39}\]2  
\[\frac{y_{n+1}-y_{n}}{\Delta t}=\frac{1}{6}[k_{1}+2k_{2}+2k_{3}+k_{4}], \tag{39}\]3  
Computation of the right-hand-side corrections \(\mathbf{r}_{1},\ldots,\mathbf{r}_{4}\) exactly requires solving the mass balance equation at all the sub-steps.