Approximation Schemes for convective term - structured grids - Schemes

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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.

Chakravarthy-Osher limiter

Sweby $\Phi$ - 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

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

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)

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)

GAMMA

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

Return to Numerical Methods

Return to Approximation Schemes for convective term - structured grids

@@ 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,
-: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,
-: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 ==

Approximation Schemes for convective term - structured grids - Schemes

From CFD-Wiki

Revision as of 20:24, 14 October 2005

Contents

reference shablon

Chakravarthy-Osher limiter

Sweby $\Phi$ - limiter

Superbee limiter

R-k limiter

MINMOD - MINimum MODulus

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

ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars

COPLA - COmbination of Piecewise Linear Approximation

HLPA - Hybrid Linear / Parabolic Approximation

CLAM - Curved-Line Advection Method

van Leer harmonic

BSOU

MSOU - Monotonic Second Order Upwind Differencing Scheme

Koren

H-CUS

MLU

SHARP - Simple High Accuracy Resolution Program

LPPA - Linear and Piecewise / Parabolic Approximasion

GAMMA

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

Views

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wiki navigation

wiki search

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Approximation Schemes for convective term - structured grids - Schemes

From CFD-Wiki

Revision as of 20:24, 14 October 2005

Contents

reference shablon

Chakravarthy-Osher limiter

Sweby - limiter

Superbee limiter

R-k limiter

MINMOD - MINimum MODulus

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

ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars

COPLA - COmbination of Piecewise Linear Approximation

HLPA - Hybrid Linear / Parabolic Approximation

CLAM - Curved-Line Advection Method

van Leer harmonic

BSOU

MSOU - Monotonic Second Order Upwind Differencing Scheme

Koren

H-CUS

MLU

SHARP - Simple High Accuracy Resolution Program

LPPA - Linear and Piecewise / Parabolic Approximasion

GAMMA

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

Views

My wiki

wiki navigation

wiki search

wiki toolbox

Sweby $\Phi$ - limiter