# Non-Linear QUICK based Schemes - structured grids

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- normalised variables + normalised variables (uniform grids)

## Contents

### 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)

 $\hat{\phi_{f}}= \begin{cases} 3 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{6} \\ \frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{1}{6} \leq \hat{\phi_{C}} \leq \frac{5}{6} \\ 1 & \frac{5}{6} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids (NVSF)

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

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $b_{f}= \left( y_{Q} - 3x_{Q}y_{Q} + 2 y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)$ (2)
 $c_{f}= \left( x_{Q}y_{Q}- y^{2}_{Q} \right)/\left( 1 - x_{Q} \right)$ (2)
 $d_{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)$ (2)
 $\boldsymbol{x_{1}=x_{Q}/3 }$ (2)
 $x_{2}= x_{Q} \left( 1 + x_{Q} - x_{Q} \right) / y_{Q}$ (2)

### 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

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

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f}+ b_{f} \hat{\phi}_{C} + c_{f} \hat{\phi}^{2}_{C} + d_{f} \hat{\phi}^{3}_{C} & 0 \leq \hat{\phi}_{C} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $b_{f}= \left[ x^{4}_{Q} + s_{Q} \left( x^{3}_{Q} - x^{2}_{Q} \right) + y_{Q} \left( 2 x_{Q} -3 x^{2}_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2$ (2)

 $c_{f}= \left[ - 2 x^{3}_{Q} + s_{Q} \left( x_{Q} - x^{3}_{Q} \right) + y_{Q} \left( 3 x^{2}_{Q} - 1 \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2$ (2)

 $d_{f} = \left[ x^{2}_{Q} + s_{Q} \left( x^{2}_{Q} - x_{Q} \right) + y_{Q} \left( 1 - 2 x_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2$ (2)

### 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

 $\hat{\phi_{f}}= \begin{cases} 2 \widehat{\phi_{C}} & 0 \leq \widehat{\phi_{C}} \leq \frac{3}{10} \\ \frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{3}{10}\leq \widehat{\phi_{C}} \leq \frac{5}{6} \\ 1 & \frac{5}{6} \leq \widehat{\phi_{C}} \leq 1 \\ \widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

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

where

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

### 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

 $\hat{\phi}_{f}= \begin{cases} 3 \hat{\phi}_{C} & 0 \leq \hat{\phi}_{C} \leq \frac{1}{6} \\ \frac{3}{8} + \frac{3}{4} \hat{\phi}_{C} & \frac{1}{6}\leq \hat{\phi}_{C} \leq \frac{1}{2} \\ \frac{3}{2} \hat{\phi_{C}} & \frac{1}{2}\leq \hat{\phi}_{C} \leq \frac{2}{3} \\ 1 & \frac{2}{3} \leq \widehat{\phi_{C}} \leq 1 \\ \hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

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

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $b_{f}= \left( y_{Q} - 3 x_{Q}y_{Q}+ 2 y^{2}_{Q} \right)/\left( x_{Q} - x^{2}_{Q} \right)$ (2)
 $c_{f}= \left( y^{2}_{Q} - x_{Q}y_{Q} \right)/\left( 1 - x_{Q} \right)$ (2)
 $d_{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)$ (2)
 $\boldsymbol{e_{f}= 0}$ (2)
 $\boldsymbol{ \hat{f}_{f} = y_{Q}/x_{Q} }$ (2)

 $\boldsymbol{ x_{1}= x_{Q}/3 }$ (2)
 $\boldsymbol{ x_{2}= x_{Q}/y_{Q} }$ (2)

### 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

 ${\phi_{f}}= {\phi}_{C} + \gamma \left( {\phi}_{C} - {\phi}_{i-1} \right) \left( \hat{\phi}^{2}_{C} - 2.5 \hat{\phi}_{C} + 1.5 \right)$ (2)
 $\gamma = \begin{cases} 1, & \left| \hat{\phi}_{C} - 1.5 \right| \leq 0.5 \\ 0, & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

normalised variables (uniform grids)

 $\hat{\phi}_f = \begin{cases} a_{f} + b_{f}\hat{\phi}_C + c_{f}\hat{\phi}^{2}_{C} + d_{f}\hat{\phi}^{3}_{C} & 0 \leq \hat{\phi}_{C} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $\boldsymbol{b_{f}= 2.5}$ (2)
 $\boldsymbol{c_{f}= - 2.5}$ (2)
 $\boldsymbol{d_{f}= 1.0 }$ (2)

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)