# A Semi-Explicit Multi-Step Method for Solving Incompressible Navier-Stokes Equations
## 3 Numerical Example
### Accuracy of the Scheme  
In this benchmark, the time accuracy and computational efficiency of the present model are tested. This benchmark was originally proposed in [10], where it was used for comparing the accuracy and stability of different fractional step methods. In this example, one solves Navier-Stokes equations in a unit square domain with homogeneous velocity conditions. A force vector is prescribed that corresponds to the following analytic solution:  
\[ \begin{split} u_{x}(x,y,t)=f(x)f^{\prime}(y)g(t),\\ u_{y}(x,y,t)=-f^{\prime}(x)f(y)g(t).\end{split}  \tag{71}\]  
where \(f\) and \(g\) are:  
\[ \begin{split} f(x)=100x^{2}(1-x)^{2},\\ g(t)=cos(4\pi t)e^{-t}.\end{split}  \tag{72}\]  
Two pressure solutions satisfy the problem: \(p=0\) and \(p=100x^{2}\). We adopt the second one as the initial condition for the pressure. The physical properties of the fluid are as follows: kinematic viscosity is 0.001 m\({}^{2}\)/s, and the density is 1 kg/m\({}^{3}\). The time interval considered was 1 s. A structured mesh of 80 \(\times\) 80 linear triangular elements was used if not mentioned otherwise. Different time step sizes were tested. The method proposed here was compared with an implicit method obtained by applying the fractional step to equations obtained using the second order backward differentiation formula (BDF2) time integration scheme (the implicit implementation was validated in [31]). Since the method proposed here is a priori expected to exhibit second order convergence, the second order implicit scheme is selected for comparison. We note that not only the rate of convergence is of importance, but also the error itself (i.e., the ordinate position of the error vs. step size curve). The velocity field at \(t=1.0\) is shown in Figure 1. The evolution of the horizontal velocity component \(U_{x}\) at the point (\(x,y\)) = (0.75, 0.75) obtained using the implicit and the present semi-explicit scheme is shown in Figures 2 and 3, respectively. One can see that as the time step diminishes, both the proposed method and the fractional step method tend to the exact solution. However, for the large time step, the implicit scheme clearly exhibits large error, while the semi-explicit Runge-Kutta scheme results in a solution nearly coinciding with the analytic one. For assessing the accuracy quantitatively, the convergence of the time approximation is displayed in Figure 4. In order to obtain the graphs, the test was performed using various time step sizes in the range between \(\Delta t=0.05\) s and \(\Delta t=0.001\) s. Error versus time step size was plotted for the proposed semi-explicit and for the implicit method. The logarithmic scale was used. The error was computed as the sum of the nodal errors at time \(t=1\) s: \(Err=\sum_{t}\frac{|\mathbf{u}_{E}-\mathbf{u}_{exact}|}{|\mathbf{u}_{exact}|}\) (n is the number of nodes). One can see that both schemes exhibit quadratic convergence. However, the error introduced by the proposed method is considerably smaller than that of the implicit method for a given time step. For example, the semi-explicit Runge-Kutta scheme with \(\Delta t=0.01\) leads to the same error as the implicit scheme with \(\Delta t=0.001\). This indicates that provided s lower computational cost (per time step) of the semi-explicit scheme, a considerable gain in the overall computational time is expected. Figure 4: Error at \(t=1\) s against the time step, logarithmic scale. Figure 3: Temporal evolution of the horizontal velocity at \((x,y)=(0.75,0.75)\). Semi-explicit Runge–Kutta scheme. Figure 2: Temporal evolution of the horizontal velocity at \((x,y)=(0.75,0.75)\). Implicit second order backward differentiation formula (BDF2) scheme. The test was performed on an Intel i5 desktop computer. It was executed on a single thread in order to gain the insight of the efficiency of the scheme without considering the gains obtained by parallel implementation.