# Derivation of New Staggered Compact Schemes with Application to Navier-Stokes Equations
## 2 Problem Formulation and Derivation of New Compact Schemes
### Higher-Order Schemes  
Higher-order schemes can be obtained in a similar way as has been done for the fourth-order case by considering higher-degree polynomial interpolations. The starting set of data has to be larger and, in order to save accuracy, derivatives employed in the Hermite interpolation have to be calculated with higher precision. By limiting ourselves to tridiagonal systems, sixth- and eighth-order basic set of derivatives can be computed through the schemes 6 CC-D1 and 8 CC-D1 (extensions of the classical Pade scheme, 4 CC-D1),\[\frac{1}{5}f^{\prime}_{j-1}+\frac{3}{5}f^{\prime}_{j}+\frac{1}{5}f^{ \prime}_{j+1} =\frac{14}{15}\frac{f_{j+1}-f_{j-1}}{2h}+\frac{1}{15}\frac{f_{j+2}-f_{j-2}}{ 4h}, \tag{13}\] \[\frac{3}{14}f^{\prime}_{j-1}+\frac{4}{7}f^{\prime}_{j}+\frac{3}{1 4}f^{\prime}_{j+1} =\frac{25}{28}\frac{f_{j+1}-f_{j-1}}{2h}+\frac{4}{35}\frac{f_{j+2} -f_{j-2}}{4h}-\frac{1}{140}\frac{f_{j+3}-f_{j-3}}{6h}, \tag{14}\]  
with truncation errors \((1/2100)h^{6}f^{VIII}\) and \((1/17,640)h^{8}f^{IX}\) respectively. The numerical scheme for the staggered first derivatives \(f^{\prime}_{j+1/2}\), can be evaluated as a linear combination of nodal first derivatives \(f^{\prime}_{k}\) for \(k=j,j+1\) and of nodal values \(f_{k}\) with \(k=j-1,\ldots j+2\), in the case of the sixth-order scheme 6 SH-D1, and with \(k=j-2,\ldots j+3\) in the case of the eighth-order scheme 8 SH-D1. For a uniform mesh these formulae read  
\[f^{\prime}_{j+1/2} =\frac{99}{64}\frac{f_{j+1}-f_{j}}{h}+\frac{1}{64}\frac{f_{j+2}-f_ {j-1}}{3h}-\frac{9}{32}\left(f^{\prime}_{j}+f^{\prime}_{j+1}\right), \tag{15}\] \[f^{\prime}_{j+1/2} =\frac{25}{16}\frac{f_{j+1}-f_{j}}{h}+\frac{25}{1024}\frac{f_{j+2 }-f_{j-1}}{3h}-\frac{1}{1024}\frac{f_{j+3}-f_{j-2}}{5h}-\frac{75}{256}\left(f^ {\prime}_{j}+f^{\prime}_{j+1}\right). \tag{16}\]  
The combination of Equations (13) and (15) gives a sixth-order scheme for the staggered first derivative at interior points, while the combination of Equations (14) and (16) gives an eighth-order scheme. The references for comparison are the classical sixth- and eighth- order staggered compact schemes, 6 SC-D1 and 8 SC-D1:  
\[\frac{9}{80}f^{\prime}_{j-1/2}+\frac{62}{80}f^{\prime}_{j+1/2}+ \frac{9}{80}f^{\prime}_{j+3/2} =\frac{63}{80}\frac{f_{j+1}-f_{j}}{h}+\frac{17}{80}\frac{f_{j+2}-f _{j-1}}{3h}, \tag{17}\] \[\frac{25}{168}f^{\prime}_{j-1/2}+\frac{59}{84}f^{\prime}_{j+1/2}+ \frac{25}{168}f^{\prime}_{j+3/2} =\frac{2675}{4032}\frac{f_{j+1}-f_{j}}{h}+\frac{925}{2688}\frac{f _{j+2}-f_{j-1}}{3h}-\frac{61}{8064}\frac{f_{j+3}-f_{j-2}}{5h}. \tag{18}\]  
The same approach can be used to determine new schemes for interpolants and second derivatives, which are based on collocated schemes for first derivatives. Specifically, the sixth-order scheme for Hermitian interpolation, 6 SH-D0, has been already presented in Equation (8) whereas the eighth-order one, 8 SH-D0, is given by the following formula,  
\[f_{j+1/2}=\frac{125}{128}\frac{f_{j+1}+f_{j}}{2}+\frac{25}{1024}\frac{f_{j+2}+ f_{j-1}}{2}-\frac{1}{1024}\frac{f_{j+3}+f_{j-2}}{2}-\frac{75h}{512}\left(f^{ \prime}_{j+1}-f^{\prime}_{j}\right). \tag{19}\]  
Equations (8) and (19) can be combined with Equations (13) and (14), thus producing a sixth- and a eighth-order scheme for the interpolant at interior points.