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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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- |
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- | == Sweby <math>\Phi</math> - limiter ==
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- |
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- | == Superbee limiter ==
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- |
| |
- | == R-k limiter ==
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- |
| |
- | == MINMOD - MINimum MODulus ==
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- |
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- | '''Harten A.''' High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393
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- |
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- | A. Harten
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- |
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- | High Resolution Schemes for Hyperbolic Conservation Laws
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- |
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- | J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991
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- |
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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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- |
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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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- |
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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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- |
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- |
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- | [[Image:NM_convectionschemes_struct_grids_Schemes_SOUCUP_Probe_01.jpg]]
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- |
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- | Normalized variables - uniform grids
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- |
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- | <table width="100%"><tr><td>
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- | :<math>
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- | \hat{\phi_{f}}=
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- | \begin{cases}
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- | \frac{3}{2} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{2} \\
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- | \frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{2} \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}
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- | </math>
| |
- | </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{\phi_{f}}=
| |
- | \begin{cases}
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- | a_{f}+ b_{f} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q} \\
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- | 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}
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- | </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>
| |
- | </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>
| |
- | </td><td width="5%">(2)</td></tr></table>
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- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
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- | c_{f}= \left( x_{Q} - y_{Q} \right)/\left( 1 - x_{Q} \right)
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
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- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
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- | d_{f} = \left( 1 - y_{Q} \right) / \left( 1 - x_{Q} \right)
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | == 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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- |
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- |
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- |
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- |
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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
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- |
| |
- | Numerical Heat Transfer, Part B, 28:23-38, 1995
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- |
| |
- | == HLPA - Hybrid Linear / Parabolic Approximation ==
| |
- |
| |
- | '''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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- |
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- | -----------------------------------------------------------------
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- |
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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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- |
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- | Usual variables
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- |
| |
- | <table width="100%"><tr><td>
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- | :<math>
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- | 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 \\
| |
- | f_{W} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1
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- | \end{cases}
| |
- | </math>
| |
- | </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>
| |
- | :<math>
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- | \hat{f_{w}}=
| |
- | \begin{cases}
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- | \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}
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- | </math>
| |
- | </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>
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- | \hat{f_{w}}=
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- | \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 \\
| |
- | \hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1
| |
- | \end{cases}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | where
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | 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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- |
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- |
| |
- | c_{w} = \left(y_{Q}- x_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) ,
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- |
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | --------------------------------------------------------
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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>
| |
- | :<math>
| |
- | \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
| |
- | \end{cases}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | for <math>\boldsymbol{U_w \triangleleft 0}</math>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \alpha^{-}_{w} =
| |
- | \begin{cases}
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- | 1 & \ if \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\
| |
- | 0 & otherwise
| |
- | \end{cases}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | and taken all the possible flow directions into account, the un-normalized form of equation can be written as
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | where
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \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>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | 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>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | [[Image: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
| |
- |
| |
- | == 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
| |
- |
| |
- | <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>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | == 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
| |
- |
| |
- |
| |
- | ----
| |
- | <i> Return to [[Numerical methods | Numerical Methods]] </i>
| |
- |
| |
- | <i> Return to [[Approximation Schemes for convective term - structured grids]] </i>
| |