# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 5 Results
### Periodic Double Mixing Layer  
The second test is a double inviscid mixing layer in a biperiodic domain. This test is challenging for numerical schemes as it presents steep gradients and formation of smaller scales. The domain is a square region of size \(2\pi\times 2\pi\) with periodic boundary conditions. The mesh has \(80^{2}\) cells along each direction, and simulations are time-advanced by means of Runge-Kutta methods. The Hermitian schemes developed in the paper are tested and compared with classical methods. The code used for this test is based on a standard pseudo-spectral algorithm, and the finite-difference schemes are emulated by means of modified wavenumbers, including shift operators and interpolation transfer functions to cope with the staggered grid. This method is highly efficient for periodic flows and allows great flexibility in comparing several spatial schemes as it only requires plugging the modified wavenumber expression of the derivative and interpolation operators (refer to Appendix A for modified wavenumbers of Hermitian schemes). The initial flowfield is given by  
\[u=\left\{ \begin{aligned} &\tanh\left(\frac{y-\pi/2}{\delta}\right), \qquad y\leq\pi,\\ &\tanh\left(\frac{3\pi/2-y}{\delta}\right),\qquad y>\pi,\end{aligned} \right. \tag{44}\]  
\[v=\epsilon\sin(x). \tag{45}\]  
The perturbation on the \(v\)-velocity is used to promote the roll-up of the shear layer; as in [35], \(\epsilon=0.05\) is used. A single instability mode is activated by choosing \(\delta=\pi/15\). \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & \multicolumn{2}{c}{\(\mathbf{u_{1}}\)} & \multicolumn{2}{c}{\(\mathbf{u_{2}}\)} \\ \hline \(\mathbf{N}\) & **Error** & **Order** & **Error** & **Order** \\ \hline
16 & \(4.167\times 10^{-4}\) & — & \(4.683\times 10^{-4}\) & — \\
32 & \(1.812\times 10^{-5}\) & 4.523 & \(2.730\times 10^{-5}\) & 4.100 \\
64 & \(6.333\times 10^{-7}\) & 4.838 & \(1.066\times 10^{-6}\) & 4.677 \\
128 & \(2.564\times 10^{-8}\) & 4.626 & \(3.752\times 10^{-8}\) & 4.829 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Grid refinement study for Burggraf flow for \(u_{1}\) and \(u_{2}\). The values are depicted in Figure 6.  
Figure 7: Contour plots of horizontal and vertical velocity component as well as stream function for the Burggraf flow testcase. Results are shown in Figure 8 in terms of iso-contours of vorticity at \(t=8\), at well-defined shear layer roll-up. In all cases, the skew-symmetric formulation of the convective term and a RK4 scheme for time-advancement are employed, except otherwise indicated. Variables are always arranged on a staggered grid layout. The reference result here is represented by the energy-conserving spectrally-accurate computation. The lower-order explicit schemes provide smeared shear layers and remarkably noisy fields, especially in proximity of the center of the mixing layer, where high gradients are being generated. Although the flowfield improves with the canonical staggered compact method, the novel Hermitian scheme provides the smoothest results, thanks to a favorable combination of high spectral accuracy and discrete energy conservation. Further insights can be gained by examining the time evolution of kinetic energy, which is shown in Figure 9. Since the case is inviscid, kinetic energy should be ideally preserved in time and stay equal to the initial value. Clearly, any scheme in divergence form does not preserve energy, except the second-order Harlow-Welch method (see left figure). The energy-violating schemes show large oscillations, which could eventually lead to a blow up. On the other hand, all the methods using the skew-symmetric form of convection preserve kinetic energy spatially, and the residual diffusive error is due to the temporal scheme (see right figure, where only methods employing skew-symmetric form are reported). At first, it is interesting to observe that schemes of increasing spatial accuracy are associated to an increasingly more dissipative temporal error.