Other Schemes (unclassified) - structured grids

== Chakravarthy-Osher limiter ==

== Sweby <math>\Phi</math> - limiter ==

== Superbee limiter ==

== R-k limiter ==

== MINMOD - MINimum MODulus ==

'''Harten A.''' High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393

A. Harten 

High Resolution Schemes for Hyperbolic Conservation Laws

J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991

Identical to SOUCUP

Normalized variables - uniform grids 

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\frac{3}{2} \hat{\phi_{C}}               &  0          \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ 
\frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{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>

Normalized variables - non-uniform grids (NVSF)

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0  \leq \hat{\phi_{C}} \leq \hat{\xi}_C \\ 
\frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } & \hat{\xi}_C  \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>

[[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]]

== SOUCUP - Second-Order Upwind Central differnce-first order UPwind  ==

{{reference-paper | author=Zhu J. | year=1992 | title=On the higher-order bounded discretization schemes for finite volume computations of incompressible flows| rest=Computational Methods in Applied Mechanics and Engineering. 98. 345-360}}

{{reference-paper | author=J. Zhu, W.Rodi | year=1991 | title=A low dispersion and bounded convection scheme | rest= Comp. Meth. Appl. Mech.&Engng, Vol. 92, p 225 }}



Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\frac{3}{2} \hat{\phi_{C}}               &  0          \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ 
\frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{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>

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_{Q} \\ 
c_{f}+ d_{f} \hat{\phi_{C}}   & x_{Q}  \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}= y_{Q}/x_{Q} }
</math>
</td><td width="5%">(2)</td></tr></table>

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

== ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars ==

Third-order flux-limiter scheme

'''M. Zijlema''' , On the construction of a third-order accurate monotone convection scheme with application to turbulent flows in general domains. International Journal for numerical methods in fluids, 22:619-641, 1996.





== COPLA - COmbination of Piecewise Linear Approximation ==

'''Seok Ki Choi, Ho Yun Nam, Mann Cho'''

Evaluation of a High-Order Bounded Convection Scheme: Three-Dimensional Numerical Experiments

Numerical Heat Transfer, Part B, 28:23-38, 1995


<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}= 
\begin{cases}
a_{f} + b_{f} \hat{\phi}_{C}  & 0 \leq \hat{\phi}_{C} \leq 0.5 x_Q \\ 
c_{f} + d_{f} \hat{\phi}_{C}  & 0.5 x_Q \leq  \hat{\phi}_{C} \leq 1.5 x_Q\\
e_{f} + f_{f} \hat{\phi}_{C}  & 1.5 x_Q \leq \hat{\phi}_{C} \leq 1 x_Q\\
\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} = \frac{2 y_{Q} - s_{Q}x_{Q}}{x_{Q}} 
</math>
</td><td width="5%">(2)</td></tr></table>

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

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

<table width="100%"><tr><td>
:<math> 
e_{f} = \frac{3 x_{Q} - 2 y_{Q} - s_{Q}x_{Q}}{3 x_{Q} - 2} 
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
f_{f} = \frac{2 y_{Q} + 2 s_{Q} x_{Q} - 2 }{3 x_{Q} - 2} 
</math>
</td><td width="5%">(2)</td></tr></table>

and 

<table width="100%"><tr><td>
<math> 
\hat{\phi}_{C}= \hat{\phi}_{C} U^{+}_{f} + \hat{\phi}_{D} U^{-}_{f} 
</math>
</td><td width="5%">(1)</td></tr></table>

== HLPA - Hybrid Linear / Parabolic Approximation ==

'''Zhu J'''. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232.

'''Zhu J., Rodi W.''' A low dispersion and bounded discretization schemes  for finite volume computations of incompressible flows // Computational Methods for Applied Mechanics and Engineering. 1991. 92. 87-96


-----------------------------------------------------------------


In this scheme, the normalized face value is approximated by a combination of linear and parabolic charachteristics passing through the points, O, Q, and P in the NVD. It satisfies TVD condition and is second-order accurate

Usual variables 

<table width="100%"><tr><td>
:<math> 
\phi_{f}= 
\begin{cases}
\phi_{f} + \left( \phi_{P} -  \phi_{W} \right) \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq 1 \\ 
\phi_{W} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\hat{\phi_{C}} \left( 2 -  \hat{\phi_{C}} \right) \hat{\phi_{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_{C}}^{2} & 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(y_{Q}- x^{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_{Q}- x_{Q} \right) /  \left(x_{Q}- x^{2}_{Q} \right)  , 
</math>
</td><td width="5%">(2)</td></tr></table>

--------------------------------------------------------
Implementation

Using the switch factors:

for <math>\boldsymbol{U_w \geq 0}</math>

<table width="100%"><tr><td>
:<math> 
\alpha^{+}_{w} =  
\begin{cases}
1 & \ \mbox{if} \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\
0 & \mbox{otherwise} 
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

for <math>\boldsymbol{U_w \triangleleft  0}</math>

<table width="100%"><tr><td>
:<math> 
\alpha^{-}_{w} =  
\begin{cases}
1 & \ \mbox{if} \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\
0 & \mbox{otherwise} 
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

and taken all the possible flow directions into account, the un-normalized form of equation can be  written as

<table width="100%"><tr><td>
:<math> 
\phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}
</math>
</td><td width="5%">(2)</td></tr></table>
  
where

<table width="100%"><tr><td>
:<math> 
\Delta \phi_{w} = U^{+}_{w} \alpha^{+}_{w} \left( \phi_{P} - \phi_{W} \right) \frac{\phi_{W} - \phi_{WW}}{\phi_{P} - \phi_{WW}} + U^{-}_{w} \alpha^{-}_{w} \left( \phi_{W} - \phi_{P} \right) \frac{\phi_{P} - \phi_{E}}{\phi_{W} - \phi_{E}}  
</math>
</td><td width="5%">(2)</td></tr></table>

<table width="100%"><tr><td>
:<math> 
 U^{+}_{w} = 0.5 \left( 1 + \left| U_{w} \right| / U_{w} \right) \ , \ U^{-}_{w} = 1 - U^{+}_{w} \ \ \left( U_{w}\neq 0 \right)
</math>
</td><td width="5%">(2)</td></tr></table>

[[Image:NM_convectionschemes_struct_grids_Schemes_HLPA_Probe_01.jpg]]

== LODA - Local Oscillation-Damping Algorithm ==

J. Zhu and M.A. Leschziner

A local oscillation-damping algorithm for higher-order convection schemes

Comput. Methods Appl. Mech. Engnrng 67 (1988) 355-366

== CLAM - Curved-Line Advection Method ==

'''Van Leer B.''' , Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics 1974; 14:361-370

identical to HLPA

normalised variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\hat{\phi_{C}} \left( 2 -  \hat{\phi_{C}} \right) \hat{\phi_{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>

normalised variables - non-uniform grids (NVSF - compare with HLPA - here is used another variant of notation)

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f} =  
\begin{cases}
\frac{\hat{\xi}_f - \hat{\xi}^{2}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C - \frac{\hat{\xi}_f - \hat{\xi}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)}\hat{\phi}^{2}_{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>

== van Leer harmonic ==

== BSOU ==

G. Papadakis, G. Bergeles. 

A locally modified second order upwind scheme for convection terms discretization.

Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995

== MSOU - Monotonic Second Order Upwind Differencing Scheme == 

Sweby

== Koren ==

bounded CUS

B. Koren 

A robust upwind discretisation method for advection, diffusion and source terms

In: Numerical Mthods for Advection-Diffusion Problems, Ed. C.B.Vreugdenhil& B.Koren, Vieweg, Braunscheweigh, p.117, (1993)

== H-CUS ==

bounded CUS

N.P.Waterson H.Deconinck 

A unified approach to the design and application of bounded high-order convection schemes

VKI-preprint, 1995-21, (1995)

== MLU ==

B. Noll

Evaluation of a bounded high-resolution scheme for combustor flow computations

AIAA J., vol. 30, No. 1, p.64 (1992)



== LPPA - Linear and Piecewise / Parabolic Approximasion ==


Normalized variables - uniform grids

<table width="100%"><tr><td>
:<math> 
\hat{\phi_{f}}=  
\begin{cases}
\frac{9}{4}{\phi}_{C} - \frac{3}{2} {\phi}^{2}_{C} &  0 \leq \hat{\phi}_{C} \leq \frac{1}{2} \\ 
\frac{1}{4}+\frac{5}{4}{\phi}_{C}-\frac{1}{2}{\phi}^{2}_{C} & \frac{1}{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>

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} & 0    \leq \hat{\phi_{C}} \leq x_{Q} \\ 
d_{f}+ d_{f} \hat{\phi}_{C} + f_{f}\hat{\phi}^{2}_{C} & x_{Q} \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( s_{Q}x^{2}_{Q} + 2x_{Q}y_{Q} \right) /  x^{2}_{Q} 
</math>
</td><td width="5%">(2)</td></tr></table>

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

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

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

== GAMMA ==

Jasak H., Weller H.G., Gosman A.D.

High resolution NVD differencing scheme for arbitrarily unstructured meshes 

International Journal for Numerical Methods in Fluids

1999, 31: 431-449

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
\hat{\phi}_C \left[ 1 + \frac{1}{2 \beta_m } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\
\frac{1}{2}\hat{\phi}_{C} + \frac{1}{2} & \beta_m \leq \hat{\phi}_C \leq 1 \\
\hat{\phi}_C & \mbox{elsewhere}
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>


<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
\hat{\phi}_C \left[1 + \frac{1}{\beta_m} \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\ 
\frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } &  \beta_m \leq \hat{\phi}_C \leq 1 \\ 
\hat{\phi}_C & \mbox{elsewhere}
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

== CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection == 

'''M.A. Alves, P.J.Oliveira, F.T. Pinho''', A convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection // International Lournal For Numerical Methods in Fluids 2003, 41; 47-75

normalised variables - uniform grid 

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
\frac{7}{4}\hat{\phi}_{C} & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{8} \\
\frac{3}{4}\hat{\phi}_{C} + \frac{3}{8} & \frac{3}{8} \leq \hat{\phi}_C  \leq \frac{3}{4} \\ 
\frac{1}{4}\hat{\phi}_{C} + \frac{3}{4} & \frac{3}{4} \triangleleft \hat{\phi}_C \triangleleft 1 \\
\hat{\phi}_{C} & \mbox{elsewhere}
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>

normalised variables - non-uniform grid (NVSF)

<table width="100%"><tr><td>
:<math> 
\hat{\phi}_{f}=  
\begin{cases}
\left[1+\frac{\hat{\xi}_f- \hat{\xi}_C}{3\left( 1 - \hat{\xi}_C \right) } \right] \frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{4}\hat{\xi}_C \\
\frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{3}{4} \hat{\xi}_C \leq \hat{\phi}_C \leq \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C \\
\frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C  \triangleleft \hat{\phi}_C \triangleleft 1 \\
\hat{\phi}_C & \mbox{elsewhere}
\end{cases}
</math>
</td><td width="5%">(2)</td></tr></table>


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