Non-Linear QUICK based Schemes - structured grids

From CFD-Wiki

1 QUICKER - Quadratic Upwind Interpolation Extended and Revised
2 SMART - Sharp and Monotonic Algorithm for Realistic Transport (Also CCCT - Curvature-Compensated Convective Transport )
3 SMARTER - SMART Efficiently Revised
4 WACEB
5 VONOS - Variable-Order Non-Oscillatory Scheme
6 CHARM - Cubic / Parabolic High-Accuracy Resolution Method
7 UMIST - Upstream Monotonic Interpolation for Scalar Transport

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)

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

(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)

Return to Numerical Methods

Return to Approximation Schemes for convective term - structured grids

Non-Linear QUICK based Schemes - structured grids

From CFD-Wiki

Contents

QUICKER - Quadratic Upwind Interpolation Extended and Revised

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

SMARTER - SMART Efficiently Revised

WACEB

VONOS - Variable-Order Non-Oscillatory Scheme

CHARM - Cubic / Parabolic High-Accuracy Resolution Method

UMIST - Upstream Monotonic Interpolation for Scalar Transport

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