Non-Linear QUICK based Schemes - structured grids

=== QUICKER - Quadratic Upwind Interpolation Extended and Revised ===

=== SMART - Sharp and Monotonic Algorithm for Realistic Transport (Also CCCT - Curvature-Compensated Convective Transport ) ===


'''P.H.Gaskell and A.C.K. Lau''', Curvature-compensated convective transport: SMART, a new boundedness preserving transport algorithm, International J. Numer. Methods Fluids 8 (1988) 617-641




Normalized variables - uniform grids (NVD)

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
3 \hat{\phi_{C}}                         &  0          \leq \hat{\phi_{C}} \leq \frac{1}{6} \\ 
\frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{1}{6} \leq \hat{\phi_{C}} \leq \frac{5}{6} \\
1                                        & \frac{5}{6} \leq \hat{\phi_{C}} \leq 1 \\    
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - non-uniform grids (NVSF)

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
a_{f}+ b_{f} \hat{\phi_{C}}   &  0    \leq \hat{\phi_{C}} \leq x_{1} \\ 
c_{f}+ d_{f} \hat{\phi_{C}}   & x{1}  \leq \hat{\phi_{C}} \leq x_{2} \\
1                             & x{2}  \leq \hat{\phi_{C}} \leq 1 \\    
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

where

<table width="100%"><tr><td>
:<math>
\boldsymbol{a_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
b_{f}= \left( y_{Q} - 3x_{Q}y_{Q} + 2 y^{2}_{Q} \right) / \left( x_{Q} -  x^{2}_{Q} \right)
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
c_{f}= \left( x_{Q}y_{Q}- y^{2}_{Q} \right)/\left( 1 - x_{Q} \right) 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
d_{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{x_{1}=x_{Q}/3 }
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
x_{2}= x_{Q} \left( 1 + x_{Q} - x_{Q} \right) / y_{Q}
</math>
</td><td width="5%">(2)</td></tr></table>

=== SMARTER - SMART Efficiently Revised ===

J.K. Shin and Y.D. Choi

Study on the improvement of the convective differencing scheme for the high-accuracy and stable resolution of the numerical solution

Trans. KSME 16(6) (1992) 1179-1194 (in Korean)


Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\frac{5}{2} \hat{\phi} + \frac{5}{2} \hat{\phi}^{2}_{C} + \hat{\phi}^{3}_{C}  &  0          \leq \hat{\phi_{C}} \leq 1 \\ 
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>


Normalized variables - non-uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
a_{f}+ b_{f} \hat{\phi}_{C} + c_{f} \hat{\phi}^{2}_{C} + d_{f} \hat{\phi}^{3}_{C}   &  0    \leq \hat{\phi}_{C} \leq 1 \\ 
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

where

<table width="100%"><tr><td>
:<math>
\boldsymbol{a_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
b_{f}= \left[ x^{4}_{Q} + s_{Q} \left( x^{3}_{Q} - x^{2}_{Q}  \right) +  y_{Q} \left( 2 x_{Q} -3 x^{2}_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
</math>
</td><td width="5%">(2)</td></tr></table>



<table width="100%"><tr><td>
:<math> 
c_{f}= \left[ - 2 x^{3}_{Q} + s_{Q} \left( x_{Q} - x^{3}_{Q}  \right) +  y_{Q} \left( 3 x^{2}_{Q} - 1 \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
</math>
</td><td width="5%">(2)</td></tr></table>


<table width="100%"><tr><td>
:<math> 
d_{f} = \left[ x^{2}_{Q} + s_{Q} \left( x^{2}_{Q} - x_{Q}  \right) +  y_{Q} \left( 1 - 2 x_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
</math>
</td><td width="5%">(2)</td></tr></table>

=== WACEB ===

Song B., Liu G.B., Kam K.Y., Amano R.S.

On a higher-order bounded discretization schemes 

International Journal for Numerical Methods in Fluids, 2000, 32, 881-897




Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
2 \widehat{\phi_{C}}                    &  0          \leq \widehat{\phi_{C}} \leq \frac{3}{10} \\ 
\frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{3}{10}\leq \widehat{\phi_{C}} \leq \frac{5}{6} \\
1                                        & \frac{5}{6} \leq \widehat{\phi_{C}} \leq 1 \\    
\widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - non-uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
a_{f}+ b_{f} \hat{\phi_{C}}   &  0    \leq \hat{\phi_{C}} \leq x_{1} \\ 
c_{f}+ d_{f} \hat{\phi_{C}}   & x_{1}  \leq \hat{\phi_{C}} \leq x_{2} \\
1                             & x_{2}  \leq \hat{\phi_{C}} \leq 1 \\    
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

where

<table width="100%"><tr><td>
:<math>
\boldsymbol{a_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{b_{f}= 2}
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
c_{f}= \left( y^{2}_{Q} - x_{Q}y_{Q} \right)/\left( 1 - x_{Q} \right) 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
d_{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
x_{1}=x_{Q}y_{Q} \left( y_{Q} - x_{Q} \right)/ \left[ 2 x_{Q} \left( 1 - x_{Q} \right) - y_{Q} \left( 1 - y_{Q} \right) \right]  
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
x_{2}= x_{Q} \left( 1 - x_{Q} + y_{Q} \right) / y_{Q}
</math>
</td><td width="5%">(2)</td></tr></table>

[[Image:NM_convectionschemes_struct_grids_Schemes_WACEB_Probe_01.jpg]]

=== VONOS - Variable-Order Non-Oscillatory Scheme ===

'''Varonos A., Bergeles G.''', Development and assessment of a Variable-Order Non-oscillatory Scheme for convection term discretization // International Journal for Numerical Methods in Fluids. 1998. 26, N 1. 1-16


Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
3 \hat{\phi}_{C}                    &  0              \leq \hat{\phi}_{C} \leq \frac{1}{6} \\ 
\frac{3}{8} + \frac{3}{4} \hat{\phi}_{C} & \frac{1}{6}\leq \hat{\phi}_{C} \leq \frac{1}{2} \\
\frac{3}{2} \hat{\phi_{C}}         & \frac{1}{2}\leq \hat{\phi}_{C}       \leq \frac{2}{3} \\
1                                  & \frac{2}{3} \leq \widehat{\phi_{C}} \leq 1 \\    
\hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - non-uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
a_{f}+ b_{f} \hat{\phi}_{C}   &  0    \leq \hat{\phi}_{C} \leq x_{1}  \\ 
c_{f}+ d_{f} \hat{\phi}_{C}   & x_{1} \leq \hat{\phi}_{C} \leq x_{Q} \\
e_{f}+ \hat{f}_{f}\hat{\phi}_{C}   & x_{Q}  \leq \hat{\phi_{C}} \leq x_{2} \\
1                             & x_{2}  \leq \hat{\phi_{C}} \leq 1 \\    
\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

where

<table width="100%"><tr><td>
:<math>
\boldsymbol{a_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
b_{f}= \left( y_{Q} - 3 x_{Q}y_{Q}+ 2 y^{2}_{Q} \right)/\left( x_{Q} - x^{2}_{Q} \right) 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
c_{f}= \left( y^{2}_{Q} - x_{Q}y_{Q} \right)/\left( 1 - x_{Q} \right) 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
d_{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math>
\boldsymbol{e_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math>
\boldsymbol{ \hat{f}_{f} = y_{Q}/x_{Q} }
</math>
</td><td width="5%">(2)</td></tr></table>


<table width="100%"><tr><td>
:<math> 
\boldsymbol{  x_{1}= x_{Q}/3 }
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{  x_{2}= x_{Q}/y_{Q} } 
</math>
</td><td width="5%">(2)</td></tr></table>

=== CHARM - Cubic / Parabolic High-Accuracy Resolution Method ===

'''G.Zhou''' , Numerical simulations of physical discontinuities in single and multi-fluid flows for arbitrary Mach numbers, PhD Thesis, Chalmers University of Technology, Sweden (1995)

Gang Zhou, Lars Davidson and Erik Olsson

Transonic Inviscid / Turbulent Airfoil Flow Simulations Using a Pressure Based Method with High Order Schemes

Lecture notes in Physics, No. 453, pp. 372-377, Springler-Verlag, Berlin, (1995)

usual variables

<table width="100%"><tr><td>
:<math> 
{\phi_{f}}= {\phi}_{C} + \gamma \left( {\phi}_{C} - {\phi}_{i-1}  \right) \left( \hat{\phi}^{2}_{C} - 2.5 \hat{\phi}_{C} + 1.5  \right) 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\gamma =  
\begin{cases}
1, &  \left| \hat{\phi}_{C} - 1.5  \right| \leq 0.5 \\ 
0, &  \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

normalised variables (uniform grids)

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_f = 
\begin{cases}
a_{f} + b_{f}\hat{\phi}_C + c_{f}\hat{\phi}^{2}_{C} + d_{f}\hat{\phi}^{3}_{C}  &  0          \leq \hat{\phi}_{C} \leq 1 \\
\hat{\phi_{C}} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>


where

<table width="100%"><tr><td>
:<math>
\boldsymbol{a_{f}= 0} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{b_{f}= 2.5}
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{c_{f}= - 2.5}
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
\boldsymbol{d_{f}= 1.0 }
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - non-uniform grids

unfortunately we cen't present expression on non-uniform grids because of complexity

=== UMIST - Upstream Monotonic Interpolation for Scalar Transport ===

'''F.S.Lien and M.A.Leschziner''' , Upstream Monotonic Interpolation for Scalar Transport with application to complex turbulent flows, International Journal for Numerical Methods in Fluids, Vol. 19, p.257, (1994)

[[Image:NM_convectionschemes_struct_grids_UMIST_probe_01.jpg]]
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