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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


Contents

Linear

SOU - Second Order Upwind (also LUDS or UDS-2)

S.P.Vanka ({{{year}}}), "Second-order upwind differencing ina recirculating flow", AIAA J., 25, 1435-1441.

R.F.Warming and R.M. Beam

Upwind second order difference schemes and applications in aerodynamics flows

AIAA J. 14 (1976) 1241-1249

Skew - Upwind

G.D.Raithby , Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)

QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS)

B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98

Usual variables

 
	f_{w}= \frac{3}{8}f_{P}+ \frac{3}{4}f_{W} - \frac{1}{8}f_{WW}
(2)

Normalised variables (uniform grid)

 
	\hat{f_{w}}= \frac{3}{8} + \frac{3}{4}\hat{f_{C}}
(2)

Normalised variables (non-uniform grid)

 
\hat{f_{w}}= \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{f_{W}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right]  \right\} U^{+}_{w} + 	\left\{ C_{2} \left( 1 + C_{3} \right) \hat{f_{P}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right]  \right\} U^{-}_{w}
(2)

LUS - Linear Upwind Scheme

H.C.Price, R.S. Varga and J.E.Warren , Application of oscillation matrices to diffusion-convection equations, Journal Math. and Phys., Vol. 45, p.301, (1966)

Fromm - Fromm's Upwind Scheme

CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)

In CUDS (UDS-4) for interpolation of function is used three upwind nodes and one node downstream.

usual variables

 
	f_{w}=\frac{1}{3}f_{P} + \frac{5}{6}f_{W} + \frac{1}{6}f_{WW}
(2)

normalised variables (uniform grids)

 
	\hat{f_{w}}=\frac{1}{3} + \frac{5}{6}\hat{f_{W}}
(2)

R.K. Aragval

A third-order-accurate upwind scheme for Navier-Stokes solution at high Reynolds numbers

Paper No. AIAA-81-0112, AIAA 19th Aerospace Science Meeting, St. Louis, 1982.

CUI - Cubic Upwind Interpolation

B.P. Leonard

A survey of finite differences of opinion on numerical muddling of incompressible defective confusion equation

paper in ASME, Applied Mechanics Division, Winter Annual Meeting, 1979



Non-Linear QUICK based

SMART - Sharp and Monotonic Algorithm for Realistic 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


NM convectionschemes struct grids SMART probe 01.jpg

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)

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

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

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)

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)

NM convectionschemes struct grids UMIST probe 01.jpg


Fromm based

Fromm scheme

J.E.Fromm

A method for reducing dispersion in convective difference schemes

J. Comp. Phys., Vol. 3, p.176, (1968)

MUSCL - Monotonic Upwind Scheme for Conservation Laws

Lien F.S. and Leschziner M.A. , Proc. 5th Int. IAHR Symp. on Refind Flow Modelling and Turbulence Measurements, Paris, Sept. 1993

NM_convectionschemes_struct_grids_Schemes_MUSCL_Probe_01.jpg

van Leer limiter

van Albada

Bounded Fromm

G.D. Van Albada, B.Van Leer, W.W.Roberts

A comparative study of computational methods in cosmic gas dynamics

Astron. Astrophysics, Vol. 108, p.76, 1982

OSPRE

bounded Fromm

Waterson [1995]

N.P.Waterson, H.Deconinck.

A unified approach to the design and application of bounded high-order convection schemes

In C. Taylor and P.Durbetaki, editors, Proc. Ninth Int. Conf. on Numer. Method. Laminar and turbulent Flow, pages 203-214, Pineride Press, Swansea, 1995

ULTIMATE Universal Limiter

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

NM convectionschemes struct grids MINMOD probe 01.jpg

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.


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.

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)

NM convectionschemes struct grids Schemes HLPA Probe 01.jpg

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

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

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

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