# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 3 Analysis of Novel Schemes
### Evaluation of the Computational Effort  
The computational cost involved in the classical fourth-order compact schemes, Equations (2), (10) and (12) can be estimated by counting the floating point operations needed for the evaluation of each formula on a mesh of \(N\) points. By assuming that both multiplications and sums have the same weight, Equations (2) and (10) require \(10N\) operations (\(2N\) for the evaluation of the right-hand side and \(8N\) for a standard tridiagonal solver), while Equation (12) requires \(14N\) operations. In a procedure in which all the three formulae are needed, as for the case of an incompressible staggered Navier-Stokes solver, the total cost of the fourth-order set of discretizations can be estimated to be \(34N\). If one employs the set of compact approximations given by Equations (7), (9) and (11), together with the evaluation of the standard fourth-order Pade scheme Equation (1), the total cost can be estimated to be \(27N\) operations, \(10N\) belonging to the standard Pade scheme Equation (1), \(5N\) for each of the formulae Equations (7) and (9) and \(7N\) for Equation (11). The operation-count saving is even more pronounced if one compares the proposed technique with standard formulae, Equations (30)-(32), or with optimized sixth-order schemes. In both of these last cases the computational effort required for the computation of staggered first derivatives and interpolations and of collocated second derivatives can be estimated to be \(42N\) operations, \(13N\) belonging to the schemes Equations (30) and (31) and \(16N\) belonging to Equation (32).  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Curve Label** & **Figure 4a** & **Figure 4b** & **Figure 4c** \\ \hline a & 4 CE-D1 & 4 SE-D0 & 4 CE-D2 \\ b & 4 CC-D1 (1) & & 4 CC-D2 (12) \\ c & 4 SC-D1 (2) & 4 SC-D0 (10) & \\ d & 6 SC-D1 (17) & 6 SC-D0 (20) & 5 CC-D2 (24) \\ e & 8 SC-D1 (18) & 8 SC-D0 (21) & 7 CC-D2 (25) \\ \hline f & 4 SH-D1 (7) & 4 SH-D0 (9) & 4 CH-D2 (11) \\ g & 6 SH-D1 (15) & 6 SH-D0 (8) & 6 CH-D2 (22) \\ h & 8 SH-D1 (16) & 8 SH-D0 (19) & 8 CH-D2 (23) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Legends corresponding to Figure 4, with letters a–e referring to classical schemes (dashed lines in Figure 4), whereas f–g to novel schemes (solid lines in the same figure). On the side of each label representing a scheme, the corresponding equation is referenced.  
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{3}{c}{_e_—**Equation (33)**} & \multicolumn{3}{c}{_e_**—**Equation (34)**} \\ \cline{2-5}  & \(\epsilon=0.1\) & \(\epsilon=0.01\) & \(\epsilon=0.001\) & \(k=1\) & \(k=2\) \\ \hline
4 CE-D1 & 0.444 & 0.240 & 0.133 & 0.540 & 0.135 \\
4 CC-D1 (Equation (1)) & 0.594 & 0.355 & 0.205 & 0.668 & 0.402 \\
4 SC-D1 (Equation (2)) & 0.782 & 0.432 & 0.243 & 0.915 & 0.965 \\
6 SC-D1 (Equation (17)) & 0.902 & 0.612 & 0.421 & 0.954 & 0.986 \\
8 SC-D1 (Equation (18)) & 0.950 & 0.702 & 0.530 & 0.969 & 0.993 \\
4 SH-D1 (Equations (1) and (7)) & 1.000 & 0.468 & 0.260 & 0.977 & 0.998 \\
6 SH-D1 (Equations (13) and (15)) & 1.000 & 0.601 & 0.405 & 0.981 & 0.998 \\
8 SH-D1 (Equations (14) and (16)) & 1.000 & 0.678 & 0.499 & 0.985 & 0.999 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Resolving efficiency \(e\) and integral resolving efficiency _e_I for selected classical and new first derivation schemes.