# Other Schemes (unclassified) - structured grids

(Difference between revisions)
 Revision as of 19:28, 29 October 2005 (view source)Michail (Talk | contribs) (→SHARP - Simple High Accuracy Resolution Program)← Older edit Latest revision as of 21:09, 10 December 2010 (view source)Michail (Talk | contribs) (→MINMOD - MINimum MODulus) (22 intermediate revisions not shown) Line 16: Line 16: J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991 J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991 + + Identical to SOUCUP + + Normalized variables - uniform grids + +
+ :$+ \hat{\phi_{f}}= + \begin{cases} + \frac{3}{2} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ + \frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{2} \leq \hat{\phi_{C}} \leq 1 \\ + \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 + \end{cases} +$ + (2)
+ + Normalized variables - non-uniform grids (NVSF) + +
+ :$+ \hat{\phi_{f}}= + \begin{cases} + \frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0 \leq \hat{\phi_{C}} \leq \hat{\xi}_C \\ + \frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } & \hat{\xi}_C \leq \hat{\phi_{C}} \leq 1 \\ + \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 + \end{cases} +$ + (2)
[[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]] [[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]] Line 26: Line 54: - [[Image:NM_convectionschemes_struct_grids_Schemes_SOUCUP_Probe_01.jpg]] Normalized variables - uniform grids Normalized variables - uniform grids Line 97: Line 124: Numerical Heat Transfer, Part B, 28:23-38, 1995 Numerical Heat Transfer, Part B, 28:23-38, 1995 + + +
+ :$+ \hat{\phi}_{f}= + \begin{cases} + a_{f} + b_{f} \hat{\phi}_{C} & 0 \leq \hat{\phi}_{C} \leq 0.5 x_Q \\ + c_{f} + d_{f} \hat{\phi}_{C} & 0.5 x_Q \leq \hat{\phi}_{C} \leq 1.5 x_Q\\ + e_{f} + f_{f} \hat{\phi}_{C} & 1.5 x_Q \leq \hat{\phi}_{C} \leq 1 x_Q\\ + \hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 + \end{cases} +$ + (2)
+ + where + +
+ :$+ \boldsymbol{a_{f} = 0 } +$ + (2)
+ +
+ :$+ b_{f} = \frac{2 y_{Q} - s_{Q}x_{Q}}{x_{Q}} +$ + (2)
+ +
+ :$+ \boldsymbol{ c_{f} = y_{Q} - s_{Q}x_{Q} } +$ + (2)
+ +
+ :$+ \boldsymbol{ d_{f} = s_{Q} } +$ + (2)
+ +
+ :$+ e_{f} = \frac{3 x_{Q} - 2 y_{Q} - s_{Q}x_{Q}}{3 x_{Q} - 2} +$ + (2)
+ +
+ :$+ f_{f} = \frac{2 y_{Q} + 2 s_{Q} x_{Q} - 2 }{3 x_{Q} - 2} +$ + (2)
+ + and + +
+ $+ \hat{\phi}_{C}= \hat{\phi}_{C} U^{+}_{f} + \hat{\phi}_{D} U^{-}_{f} +$ + (1)
== HLPA - Hybrid Linear / Parabolic Approximation == == HLPA - Hybrid Linear / Parabolic Approximation == Line 177: Line 263: \alpha^{+}_{w} = \alpha^{+}_{w} = \begin{cases} \begin{cases} - 1 & \ if \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\ + 1 & \ \mbox{if} \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\ 0 & \mbox{otherwise} 0 & \mbox{otherwise} \end{cases} \end{cases} Line 189: Line 275: \alpha^{-}_{w} = \alpha^{-}_{w} = \begin{cases} \begin{cases} - 1 & \ if \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\ + 1 & \ \mbox{if} \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\ 0 & \mbox{otherwise} 0 & \mbox{otherwise} \end{cases} \end{cases} Line 231: Line 317: '''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 '''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 + +
+ :$+ \hat{\phi_{f}}= + \begin{cases} + \hat{\phi_{C}} \left( 2 - \hat{\phi_{C}} \right) \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq 1 \\ + \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 + \end{cases} +$ + (2)
+ + normalised variables - non-uniform grids (NVSF - compare with HLPA - here is used another variant of notation) + +
+ :$+ \hat{\phi}_{f} = + \begin{cases} + \frac{\hat{\xi}_f - \hat{\xi}^{2}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C - \frac{\hat{\xi}_f - \hat{\xi}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)}\hat{\phi}^{2}_{C} & 0 \leq \hat{\phi}_{C} \leq 1 \\ + \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 + \end{cases} +$ + (2)
== van Leer harmonic == == van Leer harmonic == Line 300: Line 411: a_{f}+ b_{f} \hat{\phi}_{C} + c_{f}\hat{\phi}^{2}_{C} & 0    \leq \hat{\phi_{C}} \leq x_{Q} \\ 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      \\ 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 + \hat{\phi_{C}} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases} \end{cases} [/itex] [/itex] Line 352: Line 463: 1999, 31: 431-449 1999, 31: 431-449 + +
+ :$+ \hat{\phi}_{f}= + \begin{cases} + \hat{\phi}_C \left[ 1 + \frac{1}{2 \beta_m } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\ + \frac{1}{2}\hat{\phi}_{C} + \frac{1}{2} & \beta_m \leq \hat{\phi}_C \leq 1 \\ + \hat{\phi}_C & \mbox{elsewhere} + \end{cases} +$ + (2)
+ + +
+ :$+ \hat{\phi}_{f}= + \begin{cases} + \hat{\phi}_C \left[1 + \frac{1}{\beta_m} \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\ + \frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } & \beta_m \leq \hat{\phi}_C \leq 1 \\ + \hat{\phi}_C & \mbox{elsewhere} + \end{cases} +$ + (2)
== CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection == == 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 '''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 + +
+ :$+ \hat{\phi}_{f}= + \begin{cases} + \frac{7}{4}\hat{\phi}_{C} & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{8} \\ + \frac{3}{4}\hat{\phi}_{C} + \frac{3}{8} & \frac{3}{8} \leq \hat{\phi}_C \leq \frac{3}{4} \\ + \frac{1}{4}\hat{\phi}_{C} + \frac{3}{4} & \frac{3}{4} \triangleleft \hat{\phi}_C \triangleleft 1 \\ + \hat{\phi}_{C} & \mbox{elsewhere} + \end{cases} +$ + (2)
+ + normalised variables - non-uniform grid (NVSF) + +
+ :$+ \hat{\phi}_{f}= + \begin{cases} + \left[1+\frac{\hat{\xi}_f- \hat{\xi}_C}{3\left( 1 - \hat{\xi}_C \right) } \right] \frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{4}\hat{\xi}_C \\ + \frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{3}{4} \hat{\xi}_C \leq \hat{\phi}_C \leq \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C \\ + \frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C \triangleleft \hat{\phi}_C \triangleleft 1 \\ + \hat{\phi}_C & \mbox{elsewhere} + \end{cases} +$ + (2)

## 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

 $\hat{\phi_{f}}= \begin{cases} \frac{3}{2} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ \frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{2} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids (NVSF)

 $\hat{\phi_{f}}= \begin{cases} \frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0 \leq \hat{\phi_{C}} \leq \hat{\xi}_C \\ \frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } & \hat{\xi}_C \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

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

Zhu J. (1992), "On the higher-order bounded discretization schemes for finite volume computations of incompressible flows", Computational Methods in Applied Mechanics and Engineering. 98. 345-360.

J. Zhu, W.Rodi (1991), "A low dispersion and bounded convection scheme", Comp. Meth. Appl. Mech.&Engng, Vol. 92, p 225.

Normalized variables - uniform grids

 $\hat{\phi_{f}}= \begin{cases} \frac{3}{2} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ \frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{2} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f}+ b_{f} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q} \\ c_{f}+ d_{f} \hat{\phi_{C}} & x_{Q} \leq \hat{\phi_{C}}\leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $\boldsymbol{b_{f}= y_{Q}/x_{Q} }$ (2)
 $c_{f}= \left( x_{Q} - y_{Q} \right)/\left( 1 - x_{Q} \right)$ (2)
 $d_{f} = \left( 1 - y_{Q} \right) / \left( 1 - x_{Q} \right)$ (2)

## 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

 $\hat{\phi}_{f}= \begin{cases} a_{f} + b_{f} \hat{\phi}_{C} & 0 \leq \hat{\phi}_{C} \leq 0.5 x_Q \\ c_{f} + d_{f} \hat{\phi}_{C} & 0.5 x_Q \leq \hat{\phi}_{C} \leq 1.5 x_Q\\ e_{f} + f_{f} \hat{\phi}_{C} & 1.5 x_Q \leq \hat{\phi}_{C} \leq 1 x_Q\\ \hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f} = 0 }$ (2)
 $b_{f} = \frac{2 y_{Q} - s_{Q}x_{Q}}{x_{Q}}$ (2)
 $\boldsymbol{ c_{f} = y_{Q} - s_{Q}x_{Q} }$ (2)
 $\boldsymbol{ d_{f} = s_{Q} }$ (2)
 $e_{f} = \frac{3 x_{Q} - 2 y_{Q} - s_{Q}x_{Q}}{3 x_{Q} - 2}$ (2)
 $f_{f} = \frac{2 y_{Q} + 2 s_{Q} x_{Q} - 2 }{3 x_{Q} - 2}$ (2)

and

 $\hat{\phi}_{C}= \hat{\phi}_{C} U^{+}_{f} + \hat{\phi}_{D} U^{-}_{f}$ (1)

## 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

 $\phi_{f}= \begin{cases} \phi_{f} + \left( \phi_{P} - \phi_{W} \right) \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq 1 \\ \phi_{W} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - uniform grids

 $\hat{\phi_{f}}= \begin{cases} \hat{\phi_{C}} \left( 2 - \hat{\phi_{C}} \right) \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f} + b_{f} \hat{\phi_{C}} + c_{f} \hat{\phi_{C}}^{2} & 0 \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f} = 0 }$ (2)
 $b_{f} = \left(y_{Q}- x^{2}_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right)$ (2)
 $c_{f} = \left(y_{Q}- x_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) ,$ (2)

Implementation

Using the switch factors:

for $\boldsymbol{U_w \geq 0}$

 $\alpha^{+}_{w} = \begin{cases} 1 & \ \mbox{if} \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\ 0 & \mbox{otherwise} \end{cases}$ (2)

for $\boldsymbol{U_w \triangleleft 0}$

 $\alpha^{-}_{w} = \begin{cases} 1 & \ \mbox{if} \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\ 0 & \mbox{otherwise} \end{cases}$ (2)

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

 $\phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}$ (2)

where

 $\Delta \phi_{w} = U^{+}_{w} \alpha^{+}_{w} \left( \phi_{P} - \phi_{W} \right) \frac{\phi_{W} - \phi_{WW}}{\phi_{P} - \phi_{WW}} + U^{-}_{w} \alpha^{-}_{w} \left( \phi_{W} - \phi_{P} \right) \frac{\phi_{P} - \phi_{E}}{\phi_{W} - \phi_{E}}$ (2)
 $U^{+}_{w} = 0.5 \left( 1 + \left| U_{w} \right| / U_{w} \right) \ , \ U^{-}_{w} = 1 - U^{+}_{w} \ \ \left( U_{w}\neq 0 \right)$ (2)

## 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

 $\hat{\phi_{f}}= \begin{cases} \hat{\phi_{C}} \left( 2 - \hat{\phi_{C}} \right) \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

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

 $\hat{\phi}_{f} = \begin{cases} \frac{\hat{\xi}_f - \hat{\xi}^{2}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C - \frac{\hat{\xi}_f - \hat{\xi}_{C}}{\hat{\xi}_C \left( 1 - \hat{\xi}_C \right)}\hat{\phi}^{2}_{C} & 0 \leq \hat{\phi}_{C} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

## BSOU

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

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

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

 $\hat{\phi_{f}}= \begin{cases} \frac{9}{4}{\phi}_{C} - \frac{3}{2} {\phi}^{2}_{C} & 0 \leq \hat{\phi}_{C} \leq \frac{1}{2} \\ \frac{1}{4}+\frac{5}{4}{\phi}_{C}-\frac{1}{2}{\phi}^{2}_{C} & \frac{1}{2} \leq \hat{\phi}_{C} \leq 1 \\ \hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f}+ b_{f} \hat{\phi}_{C} + c_{f}\hat{\phi}^{2}_{C} & 0 \leq \hat{\phi_{C}} \leq x_{Q} \\ d_{f}+ d_{f} \hat{\phi}_{C} + f_{f}\hat{\phi}^{2}_{C} & x_{Q} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $b_{f}= \left( s_{Q}x^{2}_{Q} + 2x_{Q}y_{Q} \right) / x^{2}_{Q}$ (2)
 $c_{f}= \left( s_{Q}x_{Q} - y_{Q} \right)/ x^{2}_{Q}$ (2)
 $d_{f} = \left[ x^{2}_{Q} + s_{Q} \left( x^{2}_{Q} - x_{Q} \right)+ \left( 1 - 2 x_{Q} \right) \right] / \left( 1 - x_{Q} \right)^{2}$ (2)
 $e_{f} = \left[ -2 x_{Q} + s_{Q} \left( 1 - x^{2}_{Q} \right) + 2 x_{Q} y_{Q} \right] / \left( 1 - x_{Q} \right)^{2}$ (2)
 $f_{f} = \left[ 1 + s_{Q} \left( x_{Q} - 1 \right) - 2 y_{Q} \right] / \left( 1 - x_{Q} \right)^{2}$ (2)

## 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

 $\hat{\phi}_{f}= \begin{cases} \hat{\phi}_C \left[ 1 + \frac{1}{2 \beta_m } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\ \frac{1}{2}\hat{\phi}_{C} + \frac{1}{2} & \beta_m \leq \hat{\phi}_C \leq 1 \\ \hat{\phi}_C & \mbox{elsewhere} \end{cases}$ (2)

 $\hat{\phi}_{f}= \begin{cases} \hat{\phi}_C \left[1 + \frac{1}{\beta_m} \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } \left( 1 - \hat{\phi}_C \right) \right] & 0 \triangleleft \hat{\phi}_C \triangleleft \beta_m \\ \frac{ 1 - \hat{\xi}_f }{ 1 - \hat{\xi}_C } \hat{\phi}_C + \frac{ \hat{\xi}_f - \hat{\xi}_C }{ 1 - \hat{\xi}_C } & \beta_m \leq \hat{\phi}_C \leq 1 \\ \hat{\phi}_C & \mbox{elsewhere} \end{cases}$ (2)

## 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

 $\hat{\phi}_{f}= \begin{cases} \frac{7}{4}\hat{\phi}_{C} & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{8} \\ \frac{3}{4}\hat{\phi}_{C} + \frac{3}{8} & \frac{3}{8} \leq \hat{\phi}_C \leq \frac{3}{4} \\ \frac{1}{4}\hat{\phi}_{C} + \frac{3}{4} & \frac{3}{4} \triangleleft \hat{\phi}_C \triangleleft 1 \\ \hat{\phi}_{C} & \mbox{elsewhere} \end{cases}$ (2)

normalised variables - non-uniform grid (NVSF)

 $\hat{\phi}_{f}= \begin{cases} \left[1+\frac{\hat{\xi}_f- \hat{\xi}_C}{3\left( 1 - \hat{\xi}_C \right) } \right] \frac{\hat{\xi}_f}{\hat{\xi}_C} \hat{\phi}_C & 0 \triangleleft \hat{\phi}_C \triangleleft \frac{3}{4}\hat{\xi}_C \\ \frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{3}{4} \hat{\xi}_C \leq \hat{\phi}_C \leq \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C \\ \frac{\hat{\xi}_f \left(1- \hat{\xi}_f \right)}{ \hat{\xi}_C \left( 1 - \hat{\xi}_C \right)} \hat{\phi}_C + \frac{\hat{\xi}_f \left( \hat{\xi}_f - \hat{\xi}_C \right)}{1- \hat{\xi}_C} & \frac{1 + 2 \left( \hat{\xi}_f - \hat{\xi}_C \right) }{ 2 \hat{\xi}_f - \hat{\xi}_C } \hat{\xi}_C \triangleleft \hat{\phi}_C \triangleleft 1 \\ \hat{\phi}_C & \mbox{elsewhere} \end{cases}$ (2)