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Linear Schemes - structured grids

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== Linear ==
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== Introduction ==
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=== SOU - Second Order Upwind (also LUDS or UDS-2) ===
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Linear schemes UDS, CDS and these
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== SOU - Second Order Upwind (also LUDS or UDS-2) ==
{{reference-paper | author=S.P.Vanka | title=Second-order upwind differencing ina recirculating flow | rest=AIAA J., 25, 1435-1441}}
{{reference-paper | author=S.P.Vanka | title=Second-order upwind differencing ina recirculating flow | rest=AIAA J., 25, 1435-1441}}
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{{reference-paper | author=R.F.Warming and R.M. Beam | year= 1976 | title= Upwind second order difference schemes and applications in aerodynamics flows | rest= AIAA J. 14 (1976) 1241-1249  }}
{{reference-paper | author=R.F.Warming and R.M. Beam | year= 1976 | title= Upwind second order difference schemes and applications in aerodynamics flows | rest= AIAA J. 14 (1976) 1241-1249  }}
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=== Skew - Upwind ===
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== Skew - Upwind ==
'''G.D.Raithby ''', Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)
'''G.D.Raithby ''', Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)
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=== QUICK - Quadratic Upwind Interpolation for Convective Kinematics  (also UDS-3 or QUDS) ===
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== 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
'''B.P.Leonard''', A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98
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Usual variables
Usual variables
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<table width="100%"><tr><td>
 
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:<math>
 
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f_{w}= \frac{3}{8}f_{P}+ \frac{3}{4}f_{W} - \frac{1}{8}f_{WW}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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Normalised variables (non-uniform grid)
Normalised variables (non-uniform grid)
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<table width="100%"><tr><td>
 
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:<math>
 
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\begin{matrix}
 
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\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} + \\
 
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+ & \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}
 
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\end{matrix}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
<table width="100%"><tr><td>
<table width="100%"><tr><td>
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</td><td width="5%">(2)</td></tr></table>
</td><td width="5%">(2)</td></tr></table>
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=== LUS - Linear Upwind Scheme ===
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== LUS - Linear Upwind Scheme ==
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'''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)
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{{reference-paper | author=H.C.Price, R.S. Varga and J.E.Warren | year= 1966  | title= Application of oscillation matrices to diffusion-convection equations | rest= Journal Math. and Phys., Vol. 45, p.301, (1966) }}
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=== Fromm - Fromm's Upwind Scheme ===
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<table width="100%"><tr><td>
 +
:<math>
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  \phi_{f}= \phi_{C} + 0.5 \left( \phi_{C} - \phi_{U} \right)
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</math>
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</td><td width="5%">(1)</td></tr></table>
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[[Image:NM_convectionschemes_struct_grids_Schemes_FROMM_Probe_01.jpg]]
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== Fromm - Fromm's Upwind Scheme ==
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=== CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)  ===
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== 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.
In CUDS  (UDS-4) for interpolation of function is used three upwind nodes and one node downstream.
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<table width="100%"><tr><td>
<table width="100%"><tr><td>
:<math>  
:<math>  
-
f_{w}=\frac{1}{3}f_{P} + \frac{5}{6}f_{W} + \frac{1}{6}f_{WW}
+
\phi_{w}=\frac{1}{3}\phi_{P} + \frac{5}{6}\phi_{W} + \frac{1}{6}\phi_{WW}
</math>
</math>
</td><td width="5%">(2)</td></tr></table>
</td><td width="5%">(2)</td></tr></table>
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<table width="100%"><tr><td>
<table width="100%"><tr><td>
:<math>  
:<math>  
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\hat{f_{w}}=\frac{1}{3} + \frac{5}{6}\hat{f_{W}}  
+
\hat{\phi_{w}}=\frac{1}{3} + \frac{5}{6}\hat{\phi_{W}}  
</math>
</math>
</td><td width="5%">(2)</td></tr></table>
</td><td width="5%">(2)</td></tr></table>
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Paper No. AIAA-81-0112, AIAA 19th Aerospace Science Meeting, St. Louis, 1982.
Paper No. AIAA-81-0112, AIAA 19th Aerospace Science Meeting, St. Louis, 1982.
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=== CUI - Cubic Upwind Interpolation  ===
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== CUI - Cubic Upwind Interpolation  ==
B.P. Leonard
B.P. Leonard

Latest revision as of 01:11, 9 November 2005

Contents

Introduction

Linear schemes UDS, CDS and these

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 (1976), "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


\phi_{w}= \frac{3}{8}\phi_{P}+ \frac{3}{4}\phi_{W} - \frac{1}{8}\phi_{WW}
(2)
\phi_{f}= \frac{3}{8}\phi_{D}+ \frac{3}{4}\phi_{C} - \frac{1}{8}\phi_{U}
(2)


Normalised variables (uniform grid)


\hat{\phi_{f}}= \frac{3}{8} + \frac{3}{4}\hat{\phi_{C}}
(2)


Normalised variables (non-uniform grid)

\begin{matrix} \hat{\phi_{w}} & = \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{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{\phi_{P}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right] \right\} U^{-}_{w} \end{matrix}
(2)
\begin{matrix} \hat{\phi_{f}} & = \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{C}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right] \right\} U^{+}_{f} + \\ + & \left\{ C_{2} \left( 1 + C_{3} \right) \hat{\phi_{D}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right] \right\} U^{-}_{f} \end{matrix}
(2)

LUS - Linear Upwind Scheme

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

\phi_{f}= \phi_{C} + 0.5 \left( \phi_{C} - \phi_{U} \right)
(1)

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

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

normalised variables (uniform grids)

\hat{\phi_{w}}=\frac{1}{3} + \frac{5}{6}\hat{\phi_{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


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