# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 5 Results
### Burggraf Flow  
The aim of the present section is to reliably assess the formal accuracy of the novel compact method through the simulation of analytically solved benchmarks, whose exact solution allows the rigorous computation of the error. Among the scarse class of such flows, the Burggraf flow, a special case of the more general family of lid driven cavity flows, is used in the non-periodic case [41]. Indeed, it is also known as _modified_ or _analytical_ lid driven cavity [42].  
The simulation is performed in a square domain \(\mathscr{D}=\{0\leq x_{1},x_{2}\leq 1\}\) whose lid \(x_{2}=1\) is provided with a distribution of tangential velocity \(u_{1}(x,1)=f(x_{1})=16x_{1}^{2}(x_{1}-1)^{2}\); furthermore, the vertical component of the momentum equation is forced by the source term  
\[\dot{u}_{2}=-\left\{\frac{8}{\mathrm{Re}}\left[245g+2g^{\prime\prime}h^{\prime \prime}+g^{4}h\right]-64\left[\frac{g^{\prime 2}}{2}(hh^{\prime\prime\prime}-h^{ \prime}h^{\prime\prime})-hh^{\prime}(g^{\prime}g^{\prime\prime\prime}-g^{ \prime\prime 2})\right]\right\}, \tag{40}\]  
where the two functions \(g\) and \(h\) have the following expressions  
\[g(x_{1})=\int_{0}^{x_{1}}\frac{f(x_{1})}{16}\,\mathrm{d}x_{1},\qquad\qquad h(x _{2})=x_{2}^{2}(x_{2}^{2}-1). \tag{41}\]  
The exact solution, which is independent of \(\mathrm{Re}\), has the following components,  
\[u_{1}^{\mathrm{ex}}(x_{1},x_{2}) =+8g^{\prime}h^{\prime}, \tag{42}\] \[u_{2}^{\mathrm{ex}}(x_{1},x_{2}) =-8g^{\prime\prime}h. \tag{43}\]  
The simulation is performed at \(\mathrm{Re}=100\) in a computational domain of \(N\times N\) grid cells for several resolutions, namely \(N=16,32,64,128\), and the stationary state is reached by means of the classic 4 th order Runge-Kutta, starting from the initial condition of fluid at rest. The spatial discretization is achieved by means of the fourth-order Hermitian method to compute interpolants and first and second derivatives of velocity based on the velocity components (steps a1 and b1, and computation of the viscous term), and by employing classical fourth order compact schemes to compute interpolants and derivatives of velocity products (steps a3 and b3). It is worth mentioning that the novel Hermitian approach can be used, in principle, also for steps a3 and b3; this choice, however, would be less efficient, since it would require another application of the classic collocated scheme on the products of velocity to feed the Hermitian formulae.  
The error of the computed flowfield \(\mathbf{u}\) with respect to the exact solution \(\mathbf{u}^{\mathrm{ex}}\), evaluated on the grid, is measured as the infinite norm \(\left\|\mathbf{u}-\mathbf{u}^{\mathrm{ex}}\right\|_{\infty}\), listed in Table 3, and depicted in Figure 6. The contour plots of horizontal and vertical velocity, as well as of the streamfunction are reported in Figure 7.  
Figure 6: Error on \(u_{1}\) and \(u_{2}\) as function of the number of meshpoints per direction. The values are reported in Table 3 and are relative to the Burggraf flow testcase.  
The theoretical orders of accuracy are generally well reproduced, in some cases showing weak supraconvergence, a behaviour that has been already observed in literature [43].