# Approximation Schemes for convective term - structured grids - Schemes

(Difference between revisions)
 Revision as of 20:23, 14 October 2005 (view source)Michail (Talk | contribs)← Older edit Revision as of 20:24, 14 October 2005 (view source)Michail (Talk | contribs) (→fdsg)Newer edit → Line 21: Line 21: - == fdsg == - === SHARP === - B. P. Leonard. Simple high-accuracy resolution program for convective modelling of discontinuities. - - International Journal for Numerical Methods in Fluids, 8:1291–1318, 1988. - - === ULTIMATE - Universal Limiter for Transport Interpolation Modelling of the Advective Transport Equation === - - B. P. Leonard. Universal limiter for transient interpolation modelling of the advective transport - equations. Technical Memorandum TM-100916 ICOMP-88-11, NASA, 1988. - - === ULTIMATE-QUICKEST === - - B. P. Leonard. The ULTIMATE conservative difference scheme applied to unsteady one–dimensional advection. Computer Methods in Applied Mechanics and Engineering, 88:17–74, - June 1991. - - === ULTRA-SHARP : Universal Limiter for Thight Resolution and Accuracy in combination with the Simple High-Accuracy Resolution Program (also ULTRA-QUICK) === - - B. P. Leonard and S. Mokhtari. - Beyond first-order upwinding: the ULTRA-SHARP alternative for non-oscillatory steady state simulation of convection. International Journal of Numerical - Methods in Engineering, 30:729–766, 1990. - - B. P. Leonard and S. Mokhtari. - ULTRA-SHARP nonoscillatory convection schemes for highspeed steady multidimensional flow. Technical Memorandum TM-102568 ICOMP-90-12, - NASA, April 1990. - - === UTOPIA - Uniformly Third Order Polynomial Interpolation Algorithm === - - B. P. Leonard, M. K. MacVean, and A. P. Lock. - - Positivity-preserving numerical schemes for multidimensional advection. Technical Memorandum TM-106055 ICOMP-93-05, NASA, March 1993. - - === NIRVANA - Non-oscilatory Integrally Reconstructed Volume-Avaraged Numerical Advection scheme === - - B. P. Leonard, A. P. Lock, and M. K. MacVean. - The NIRVANA scheme applied to one–dimensional advection. International Journal of Numerical Methods in Heat and Fluid Flow, - 5:341–377, 1995. - - === ENIGMATIC - Extended Numerical Integration for Genuinely Multidimensional Advective Transport Insuring Conservation === - - B. P. Leonard, A. P. Lock, and M. K. MacVean. - - Extended numerical integration for genuinely multidimensional advective transport insuring conservation. - - In C. Taylor and P. Durbetaki, editors, Numerical Methods in Laminar and Turbulent Flow, volume 9, pages 1–12. Pineridge - Press, 1995. - - === MACHO : Multidimensional Advective - Conservative Hybrid Operator === - - B. P. Leonard, A. P. Lock, and M. K. MacVean. - - Conservative explicit unrestricted-timestep multidimensional constancy-preserving advection schemes. Monthly Weather Review, - 124:2588–2606, November 1996. - - === COSMIC : Conservative Operator Splitting for Multidimensions with Internal Constancy === - - B. P. Leonard, A. P. Lock, and M. K. MacVean. Conservative explicit unrestricted-timestep multidimensional constancy-preserving advection schemes. Monthly Weather Review, 124:2588–2606, November 1996. - - === QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms === - - B. P. Leonard. Elliptic systems: Finite-difference method IV. In W. J. Minkowycz, E. M. Sparrow, G. E. Schneider, and R. H. Pletcher, editors, Handbook of Numerical Heat Transfer, pages 347–378. Wiley, New York, 1988. - - === AQUATIC - Adjusted Quadratic Upstream Algorithm for Transient Incompressible Convection === - - B. P. Leonard - A survey of finite differences with upwinding for numerical modelling of the incompressible convective diffusion equation. - - In C. Taylor and K. Morgan, editors, Computational - Methods in Transient and Turbulent Flow, pages 1–35. Pineridge Press, Swansea, 1981. - - === EXQUISITE - Exponential or Quadratic Upstream Interpolation for Solution of the Incompressible Transport Equation === - - B. P. Leonard. - - A survey of finite differences with upwinding for numerical modelling of the incompressible convective diffusion equation. - - In C. Taylor and K. Morgan, editors, Computational Methods in Transient and Turbulent Flow, pages 1–35. Pineridge Press, Swansea, 1981. == Chakravarthy-Osher limiter == == Chakravarthy-Osher limiter ==

## Revision as of 20:24, 14 October 2005

When we shall fill this page, I offer to make common identifications, because in different issues was used different notation.

Also we beg everybody to help me with original works. Later I shall write, what is necessary. If anyone have literature connected with convective schemes, please drop me a line.

We shall be very glad and grateful to hear any critical suggestion (please drop a few lines at Wiki Forum)

It is just a skeleton, but we hope that it will be developed into the good thing

## reference shablon

SMITH (3000), "XXX", XXX.

## 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

## 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

## 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

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

Normalized variables - uniform grids

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

Normalized variables - non-uniform grids

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

where

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

Implementation

Using the switch factors:

for $\boldsymbol{U_w \geq 0}$

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

for $\boldsymbol{U_w \triangleleft 0}$

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

## 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

## 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)

## SHARP - Simple High Accuracy Resolution Program

B.P.Leonard, Simple high-accuracy resolution rogram for convective modelling of discontinuities, International J. Numerical Methods Fluids 8 (1988) 1291-1381

## 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)

## 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