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Approximation Schemes for convective term - structured grids - Schemes

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{{reference-paper | author=SMITH    | year= 3000  | title= XXX
{{reference-paper | author=SMITH    | year= 3000  | title= XXX
     | rest= XXX  }}
     | rest= XXX  }}
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== Chakravarthy-Osher limiter ==
 
-
 
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== Sweby <math>\Phi</math> - limiter ==
 
-
 
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== Superbee limiter ==
 
-
 
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== R-k limiter ==
 
-
 
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== MINMOD - MINimum MODulus ==
 
-
 
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'''Harten A.''' High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393
 
-
 
-
A. Harten
 
-
 
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High Resolution Schemes for Hyperbolic Conservation Laws
 
-
 
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J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991
 
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[[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]]
 
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== SOUCUP - Second-Order Upwind Central differnce-first order UPwind  ==
 
-
 
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{{reference-paper | author=Zhu J. | year=1992 | title=On the higher-order bounded discretization schemes for finite volume computations of incompressible flows| rest=Computational Methods in Applied Mechanics and Engineering. 98. 345-360}}
 
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{{reference-paper | author=J. Zhu, W.Rodi | year=1991 | title=A low dispersion and bounded convection scheme | rest= Comp. Meth. Appl. Mech.&Engng, Vol. 92, p 225 }}
 
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[[Image:NM_convectionschemes_struct_grids_Schemes_SOUCUP_Probe_01.jpg]]
 
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Normalized variables - uniform grids
 
-
 
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<table width="100%"><tr><td>
 
-
:<math>
 
-
\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
 
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\end{cases}
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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Normalized variables - non-uniform grids
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
-
\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    \\
 
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\hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1
 
-
\end{cases}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
where
 
-
 
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<table width="100%"><tr><td>
 
-
:<math>
 
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\boldsymbol{a_{f}= 0}
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
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<table width="100%"><tr><td>
 
-
:<math>
 
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\boldsymbol{b_{f}= y_{Q}/x_{Q} }
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
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c_{f}= \left( x_{Q} - y_{Q} \right)/\left( 1 - x_{Q} \right)
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
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d_{f} = \left( 1 - y_{Q} \right) / \left( 1 - x_{Q} \right)
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
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== ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars ==
 
-
 
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Third-order flux-limiter scheme
 
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'''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.
 
-
 
-
 
-
 
-
 
-
 
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== COPLA - COmbination of Piecewise Linear Approximation ==
 
-
 
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'''Seok Ki Choi, Ho Yun Nam, Mann Cho'''
 
-
 
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Evaluation of a High-Order Bounded Convection Scheme: Three-Dimensional Numerical Experiments
 
-
 
-
Numerical Heat Transfer, Part B, 28:23-38, 1995
 
-
 
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== HLPA - Hybrid Linear / Parabolic Approximation ==
 
-
 
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'''Zhu J'''. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232.
 
-
 
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'''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
 
-
 
-
 
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-----------------------------------------------------------------
 
-
 
-
 
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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
 
-
 
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Usual variables
 
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<table width="100%"><tr><td>
 
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:<math>
 
-
f_{w}=
 
-
\begin{cases}
 
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f_{w} + \left( f_{P} -  f_{W} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\
 
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f_{W} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1
 
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\end{cases}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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Normalized variables - uniform grids
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
-
\hat{f_{w}}= 
 
-
\begin{cases}
 
-
\hat{f_{C}} \left( 2 -  \hat{f_{C}} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\
 
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\hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1
 
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\end{cases}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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Normalized variables - non-uniform grids
 
-
 
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<table width="100%"><tr><td>
 
-
:<math>
 
-
\hat{f_{w}}=
 
-
\begin{cases}
 
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a_{w} + b_{w} \hat{f_{C}} + c_{w} \hat{f_{C}}^{2} & 0 \leq \hat{f_{C}} \leq 1 \\
 
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\hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1
 
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\end{cases}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
-
where
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
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a_{w} = 0  , 
 
-
 
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b_{w} = \left(y_{Q}- x^{2}_{Q} \right) /  \left(x_{Q}- x^{2}_{Q} \right)  ,
 
-
 
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c_{w} = \left(y_{Q}- x_{Q} \right) /  \left(x_{Q}- x^{2}_{Q} \right)  ,
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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--------------------------------------------------------
 
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Implementation
 
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Using the switch factors:
 
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for <math>\boldsymbol{U_w \geq 0}</math>
 
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<table width="100%"><tr><td>
 
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:<math>
 
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\alpha^{+}_{w} = 
 
-
\begin{cases}
 
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1 & \ if \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\
 
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0 & otherwise
 
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\end{cases}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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for <math>\boldsymbol{U_w \triangleleft  0}</math>
 
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<table width="100%"><tr><td>
 
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:<math>
 
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\alpha^{-}_{w} = 
 
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\begin{cases}
 
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1 & \ if \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\
 
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0 & otherwise
 
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\end{cases}
 
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</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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and taken all the possible flow directions into account, the un-normalized form of equation can be  written as
 
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<table width="100%"><tr><td>
 
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:<math>
 
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\phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
 
-
where
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
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\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}} 
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
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<table width="100%"><tr><td>
 
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:<math>
 
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U^{+}_{w} = 0.5 \left( 1 + \left| U_{w} \right| / U_{w} \right) \ , \ U^{-}_{w} = 1 - U^{+}_{w} \ \ \left( U_{w}\neq 0 \right)
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
-
 
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[[Image:NM_convectionschemes_struct_grids_Schemes_HLPA_Probe_01.jpg]]
 
-
 
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== CLAM - Curved-Line Advection Method ==
 
-
 
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'''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
 
-
 
-
 
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== van Leer harmonic ==
 
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== BSOU ==
 
-
 
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G. Papadakis, G. Bergeles.
 
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A locally modified second order upwind scheme for convection terms discretization.
 
-
 
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Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995
 
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== MSOU - Monotonic Second Order Upwind Differencing Scheme ==
 
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Sweby
 
-
 
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== Koren ==
 
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bounded CUS
 
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B. Koren
 
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A robust upwind discretisation method for advection, diffusion and source terms
 
-
 
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In: Numerical Mthods for Advection-Diffusion Problems, Ed. C.B.Vreugdenhil& B.Koren, Vieweg, Braunscheweigh, p.117, (1993)
 
-
 
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== H-CUS ==
 
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bounded CUS
 
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N.P.Waterson H.Deconinck
 
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A unified approach to the design and application of bounded high-order convection schemes
 
-
 
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VKI-preprint, 1995-21, (1995)
 
-
 
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== MLU ==
 
-
 
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B. Noll
 
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Evaluation of a bounded high-resolution scheme for combustor flow computations
 
-
 
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AIAA J., vol. 30, No. 1, p.64 (1992)
 
-
 
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== SHARP - Simple High Accuracy Resolution Program ==
 
-
 
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'''B.P.Leonard''', Simple high-accuracy resolution rogram for convective modelling of discontinuities, International J. Numerical Methods Fluids 8 (1988) 1291-1381
 
-
 
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== LPPA - Linear and Piecewise / Parabolic Approximasion ==
 
-
 
-
 
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Normalized variables - uniform grids
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
\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}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
Normalized variables - non-uniform grids
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
\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}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
where
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
\boldsymbol{a_{f}= 0}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
b_{f}= \left( s_{Q}x^{2}_{Q} + 2x_{Q}y_{Q} \right) /  x^{2}_{Q}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
c_{f}= \left( s_{Q}x_{Q} - y_{Q} \right)/ x^{2}_{Q}
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
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} 
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
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} 
 
-
</math>
 
-
</td><td width="5%">(2)</td></tr></table>
 
-
 
-
<table width="100%"><tr><td>
 
-
:<math>
 
-
f_{f} = \left[ 1 + s_{Q} \left( x_{Q} - 1 \right) - 2 y_{Q} \right] / \left( 1 - x_{Q} \right)^{2} 
 
-
</math>
 
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</td><td width="5%">(2)</td></tr></table>
 
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-
== GAMMA ==
 
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-
== 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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----
 
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<i> Return to [[Numerical methods | Numerical Methods]] </i>
 
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-
<i> Return to [[Approximation Schemes for convective term - structured grids]] </i>
 

Revision as of 20:28, 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


  1. Linear Schemes - structured grids
  2. Non-Linear QUICK based Schemes - structured grids
  3. Fromm based Schemes - structured grids
  4. Schemes by Leonard - structured grids
  5. Other Schemes (unclassified) - structured grids

reference shablon

SMITH (3000), "XXX", XXX.

My wiki