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- | ''When we shall fill this page, I offer to make common identifications, because in different issues was used different notation. | + | ''When we shall fill this page, we offer to make common identifications and definitions, because in different issues was used different notation. |
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- | ''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.'' | + | ''Also we beg everybody to help us with original works. Please see section about what we need. If anyone have literature connected with convective schemes, please drop us a line. Of course You are welcome to participate in Wiki'' |
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| ''We shall be very glad and grateful to hear any critical suggestion (please drop a few lines at Wiki Forum)'' | | ''We shall be very glad and grateful to hear any critical suggestion (please drop a few lines at Wiki Forum)'' |
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| ''It is just a skeleton, but we hope that it will be developed into the good thing'' | | ''It is just a skeleton, but we hope that it will be developed into the good thing'' |
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- | == reference shablon == | + | == [[Approximation Schemes for convective term - structured grids - Common#Basic Discretisation schemes | Basic Discretisation schemes]] == |
| + | * [[Approximation Schemes for convective term - structured grids - Common#Central Differencing Scheme (CDS) | Central Differencing Scheme (CDS) ]] |
| + | * [[Approximation Schemes for convective term - structured grids - Common#Upwind Differencing Scheme (UDS) also (First-order upwind - FOU) | Upwind Differencing Scheme (UDS) also (First-order upwind - FOU) ]] |
| + | * [[Approximation Schemes for convective term - structured grids - Common#Hybrid Differencing Scheme (HDS also HYBRID) |Hybrid Differencing Scheme (HDS also HYBRID) ]] |
| + | * [[Approximation Schemes for convective term - structured grids - Common#Power-Law Scheme (also Exponencial scheme or PLDS )| Power-Law Scheme (also Exponencial scheme or PLDS ) ]] |
| | | |
- | {{reference-paper | author=SMITH | year= 3000 | title= XXX
| + | == [[Linear Schemes - structured grids ]] == |
- | | rest= XXX }}
| + | |
| | | |
| + | * [[Linear Schemes - structured grids #SOU - Second Order Upwind (also LUDS or UDS-2) | SOU - Second Order Upwind (also LUDS or UDS-2) ]] |
| + | * [[Linear Schemes - structured grids #Skew - Upwind |Skew - Upwind ]] |
| + | * [[Linear Schemes - structured grids #QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS) |QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS) ]] |
| + | * [[Linear Schemes - structured grids #LUS - Linear Upwind Scheme| LUS - Linear Upwind Scheme ]] |
| + | * [[Linear Schemes - structured grids #Fromm - Fromm's Upwind Scheme |Fromm - Fromm's Upwind Scheme ]] |
| + | * [[Linear Schemes - structured grids #CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4) |CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4) ]] |
| + | * [[Linear Schemes - structured grids #CUI - Cubic Upwind Interpolation |CUI - Cubic Upwind Interpolation ]] |
| | | |
- | == Linear == | + | == [[Non-Linear QUICK based Schemes - structured grids ]] == |
| | | |
- | === SOU - Second Order Upwind (also LUDS or UDS-2) ===
| + | * [[Non-Linear QUICK based Schemes - structured grids #QUICKER - Quadratic Upwind Interpolation Extended and Revised | QUICKER - Quadratic Upwind Interpolation Extended and Revised ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #SMART - Sharp and Monotonic Algorithm for Realistic Transport (Also CCCT - Curvature-Compensated Convective Transport ) |SMART - Sharp and Monotonic Algorithm for Realistic Transport (Also CCCT - Curvature-Compensated Convective Transport ) ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #SMARTER - SMART Efficiently Revised |SMARTER - SMART Efficiently Revised ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #WACEB |WACEB ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #VONOS - Variable-Order Non-Oscillatory Scheme |VONOS - Variable-Order Non-Oscillatory Scheme ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #CHARM - Cubic / Parabolic High-Accuracy Resolution Method |CHARM - Cubic / Parabolic High-Accuracy Resolution Method ]] |
| + | * [[Non-Linear QUICK based Schemes - structured grids #UMIST - Upstream Monotonic Interpolation for Scalar Transport |UMIST - Upstream Monotonic Interpolation for Scalar Transport ]] |
| | | |
- | {{reference-paper | author=S.P.Vanka | title=Second-order upwind differencing ina recirculating flow | rest=AIAA J., 25, 1435-1441}}
| + | == [[ Fromm based Schemes - structured grids ]] == |
| | | |
- | {{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 }}
| + | * [[Fromm based Schemes - structured grids#Fromm scheme| Fromm scheme]] |
| + | * [[Fromm based Schemes - structured grids#MUSCL - Monotonic Upwind Scheme for Conservation Laws |MUSCL - Monotonic Upwind Scheme for Conservation Laws ]] |
| + | * [[Fromm based Schemes - structured grids#van Leer limiter |van Leer limiter ]] |
| + | * [[Fromm based Schemes - structured grids#van Albada | van Albada ]] |
| + | * [[Fromm based Schemes - structured grids#OSPRE |OSPRE ]] |
| | | |
- | === Skew - Upwind === | + | == [[Schemes by Leonard - structured grids ]] == |
| | | |
- | '''G.D.Raithby ''', Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)
| + | * [[Schemes by Leonard - structured grids#SHARP - Simple High Accuracy Resolution Program | SHARP - Simple High Accuracy Resolution Program]] |
| + | * [[Schemes by Leonard - structured grids#ULTIMATE - Universal Limiter for Transport Interpolation Modelling of the Advective Transport Equation | ULTIMATE - Universal Limiter for Transport Interpolation Modelling of the Advective Transport Equation]] |
| + | * [[Schemes by Leonard - structured grids#ULTIMATE-QUICKEST |ULTIMATE-QUICKEST ]] |
| + | * [[Schemes by Leonard - structured grids#ULTRA-SHARP : Universal Limiter for Thight Resolution and Accuracy in combination with the Simple High-Accuracy Resolution Program (also ULTRA-QUICK) |ULTRA-SHARP : Universal Limiter for Thight Resolution and Accuracy in combination with the Simple High-Accuracy Resolution Program (also ULTRA-QUICK) ]] |
| + | * [[Schemes by Leonard - structured grids#UTOPIA - Uniformly Third Order Polynomial Interpolation Algorithm |UTOPIA - Uniformly Third Order Polynomial Interpolation Algorithm ]] |
| + | * [[Schemes by Leonard - structured grids#NIRVANA - Non-oscilatory Integrally Reconstructed Volume-Avaraged Numerical Advection scheme |NIRVANA - Non-oscilatory Integrally Reconstructed Volume-Avaraged Numerical Advection scheme ]] |
| + | * [[Schemes by Leonard - structured grids#ENIGMATIC - Extended Numerical Integration for Genuinely Multidimensional Advective Transport Insuring Conservation |ENIGMATIC - Extended Numerical Integration for Genuinely Multidimensional Advective Transport Insuring Conservation ]] |
| + | * [[Schemes by Leonard - structured grids#MACHO : Multidimensional Advective - Conservative Hybrid Operator |MACHO : Multidimensional Advective - Conservative Hybrid Operator ]] |
| + | * [[Schemes by Leonard - structured grids#COSMIC : Conservative Operator Splitting for Multidimensions with Internal Constancy |COSMIC : Conservative Operator Splitting for Multidimensions with Internal Constancy ]] |
| + | * [[Schemes by Leonard - structured grids#QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms |QUICKEST - Quadratic Upstream Interpolation for Convective Kinematics with Estimated Streaming Terms ]] |
| + | * [[Schemes by Leonard - structured grids#AQUATIC - Adjusted Quadratic Upstream Algorithm for Transient Incompressible Convection |AQUATIC - Adjusted Quadratic Upstream Algorithm for Transient Incompressible Convection ]] |
| + | * [[Schemes by Leonard - structured grids#EXQUISITE - Exponential or Quadratic Upstream Interpolation for Solution of the Incompressible Transport Equation |EXQUISITE - Exponential or Quadratic Upstream Interpolation for Solution of the Incompressible Transport Equation ]] |
| + | * [[Schemes by Leonard - structured grids#EULER-QUICK |EULER-QUICK ]] |
| | | |
- | === QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS) === | + | == [[Other Schemes (unclassified) - structured grids ]] == |
| | | |
- | '''B.P.Leonard''', A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98 | + | ''Here we placed schemes which were not included into the previous sections (or we can't subdivide its into some class)'' |
| | | |
- | Usual variables
| |
| | | |
- | <table width="100%"><tr><td>
| + | * [[Other Schemes (unclassified) - structured grids #Chakravarthy-Osher limiter |Chakravarthy-Osher limiter ]] |
- | :<math>
| + | * [[Other Schemes (unclassified) - structured grids #Sweby Φ - limiter |Sweby Φ - limiter ]] |
- | f_{w}= \frac{3}{8}f_{P}+ \frac{3}{4}f_{W} - \frac{1}{8}f_{WW}
| + | * [[Other Schemes (unclassified) - structured grids #Superbee limiter |Superbee limiter ]] |
- | </math>
| + | * [[Other Schemes (unclassified) - structured grids #R-k limiter |R-k limiter ]] |
- | </td><td width="5%">(2)</td></tr></table>
| + | * [[Other Schemes (unclassified) - structured grids #MINMOD - MINimum MODulus |MINMOD - MINimum MODulus ]] |
| + | * [[Other Schemes (unclassified) - structured grids #SOUCUP - Second-Order Upwind Central differnce-first order UPwind |SOUCUP - Second-Order Upwind Central differnce-first order UPwind ]] |
| + | * [[Other Schemes (unclassified) - structured grids #ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars |ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars ]] |
| + | * [[Other Schemes (unclassified) - structured grids #COPLA - COmbination of Piecewise Linear Approximation |COPLA - COmbination of Piecewise Linear Approximation ]] |
| + | * [[Other Schemes (unclassified) - structured grids #HLPA - Hybrid Linear / Parabolic Approximation |HLPA - Hybrid Linear / Parabolic Approximation ]] |
| + | * [[Other Schemes (unclassified) - structured grids #LODA - Local Oscillation-Damping Algorithm |LODA - Local Oscillation-Damping Algorithm ]] |
| + | * [[Other Schemes (unclassified) - structured grids #CLAM - Curved-Line Advection Method |CLAM - Curved-Line Advection Method ]] |
| + | * [[Other Schemes (unclassified) - structured grids #van Leer harmonic |van Leer harmonic ]] |
| + | * [[Other Schemes (unclassified) - structured grids #BSOU |BSOU ]] |
| + | * [[Other Schemes (unclassified) - structured grids #MSOU - Monotonic Second Order Upwind Differencing Scheme |MSOU - Monotonic Second Order Upwind Differencing Scheme ]] |
| + | * [[Other Schemes (unclassified) - structured grids #Koren |Koren ]] |
| + | * [[Other Schemes (unclassified) - structured grids #H-CUS |H-CUS ]] |
| + | * [[Other Schemes (unclassified) - structured grids #MLU |MLU ]] |
| + | * [[Other Schemes (unclassified) - structured grids #LPPA - Linear and Piecewise / Parabolic Approximasion |LPPA - Linear and Piecewise / Parabolic Approximasion ]] |
| + | * [[Other Schemes (unclassified) - structured grids #GAMMA |GAMMA ]] |
| + | * [[Other Schemes (unclassified) - structured grids #CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection |CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection ]] |
| | | |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \phi_{w}= \frac{3}{8}\phi_{P}+ \frac{3}{4}\phi_{W} - \frac{1}{8}\phi_{WW}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \phi_{f}= \frac{3}{8}\phi_{D}+ \frac{3}{4}\phi_{C} - \frac{1}{8}\phi_{U}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- |
| |
- | Normalised variables (uniform grid)
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \hat{f_{w}}= \frac{3}{8} + \frac{3}{4}\hat{f_{C}}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \hat{\phi_{w}}= \frac{3}{8} + \frac{3}{4}\hat{\phi_{C}}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \hat{\phi_{f}}= \frac{3}{8} + \frac{3}{4}\hat{\phi_{C}}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- |
| |
- | Normalised variables (non-uniform grid)
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \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}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \begin{matrix}
| |
- | \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}
| |
- | \end{matrix}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
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- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \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}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \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}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | === 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 ===
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- |
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- |
| |
- | [[Image:NM_convectionschemes_struct_grids_Schemes_FROMM_Probe_01.jpg]]
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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.
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- |
| |
- | usual variables
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- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | f_{w}=\frac{1}{3}f_{P} + \frac{5}{6}f_{W} + \frac{1}{6}f_{WW}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
| |
- | normalised variables (uniform grids)
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- |
| |
- | <table width="100%"><tr><td>
| |
- | :<math>
| |
- | \hat{f_{w}}=\frac{1}{3} + \frac{5}{6}\hat{f_{W}}
| |
- | </math>
| |
- | </td><td width="5%">(2)</td></tr></table>
| |
- |
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- | R.K. Aragval
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- |
| |
- | A third-order-accurate upwind scheme for Navier-Stokes solution at high Reynolds numbers
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- |
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- | 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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- |
| |
- | B.P. Leonard
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- |
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- | A survey of finite differences of opinion on numerical muddling of incompressible defective confusion equation
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- |
| |
- | paper in ASME, Applied Mechanics Division, Winter Annual Meeting, 1979
| |
| | | |
| | | |
| --------------------------------------------------------------------- | | --------------------------------------------------------------------- |
| | | |
- | == Non-Linear QUICK based == | + | == reference shablon == |
| | | |
- | === SMART - Sharp and Monotonic Algorithm for Realistic Transport ===
| + | {{reference-paper | author=SMITH | year= 3000 | title= XXX |
- | | + | | rest= XXX }} |
- | '''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
| + | |
- | | + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_SMART_probe_01.jpg]]
| + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_Schemes_SMART_Probe_01.jpg]]
| + | |
- | | + | |
- | | + | |
- | Normalized variables - uniform grids
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \hat{\phi_{f}}=
| + | |
- | \begin{cases}
| + | |
- | 3 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{6} \\
| + | |
- | \frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{1}{6} \leq \hat{\phi_{C}} \leq \frac{5}{6} \\
| + | |
- | 1 & \frac{5}{6} \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_{w}+ b_{w} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{1} \\
| + | |
- | c_{w}+ d_{w} \hat{\phi_{C}} & x{1} \leq \hat{\phi_{C}} \leq x_{2} \\
| + | |
- | 1 & x{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>
| + | |
- | | + | |
- | 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( y_{Q} - 3x_{Q}y_{Q} + 2 y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | c_{f}= \left( x_{Q}y_{Q}- y^{2}_{Q} \right)/\left( 1 - x_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | d{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \boldsymbol{x_{1}=x_{Q}/3 }
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | x_{2}= x_{Q} \left( 1 + x_{Q} - x_{Q} \right) / y_{Q}
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | === 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)
| + | |
- | | + | |
- | | + | |
- | Normalized variables - uniform grids
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \hat{\phi_{f}}=
| + | |
- | \begin{cases}
| + | |
- | \frac{5}{2} \hat{\phi} + \frac{5}{2} \hat{\phi}^{2}_{C} + \hat{\phi}^{3}_{C} & 0 \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} + d_{f} \hat{\phi}^{3}_{C} & 0 \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[ x^{4}_{Q} + s_{Q} \left( x^{3}_{Q} - x^{2}_{Q} \right) + y_{Q} \left( 2 x_{Q} -3 x^{2}_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | | + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | c_{f}= \left[ - 2 x^{3}_{Q} + s_{Q} \left( x_{Q} - x^{3}_{Q} \right) + y_{Q} \left( 3 x^{2}_{Q} - 1 \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
| + | |
- | </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) + y_{Q} \left( 1 - 2 x_{Q} \right) \right] / \left( x_{Q} - x^{2}_{Q} \right)^2
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | === 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
| + | |
- | | + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_Schemes_WACEB_Probe_01.jpg]]
| + | |
- | | + | |
- | | + | |
- | Normalized variables - uniform grids
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \hat{\phi_{f}}=
| + | |
- | \begin{cases}
| + | |
- | 2 \widehat{\phi_{C}} & 0 \leq \widehat{\phi_{C}} \leq \frac{3}{10} \\
| + | |
- | \frac{3}{8} + \frac{3}{4} \hat{\phi_{C}} & \frac{3}{10}\leq \widehat{\phi_{C}} \leq \frac{5}{6} \\
| + | |
- | 1 & \frac{5}{6} \leq \widehat{\phi_{C}} \leq 1 \\
| + | |
- | \widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\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}} & 0 \leq \hat{\phi_{C}} \leq x_{1} \\
| + | |
- | c_{f}+ d_{f} \hat{\phi_{C}} & x_{1} \leq \hat{\phi_{C}} \leq x_{2} \\
| + | |
- | 1 & x_{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>
| + | |
- | | + | |
- | 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>
| + | |
- | \boldsymbol{b_{f}= 2}
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | c_{f}= \left( y^{2}_{Q} - x_{Q}y_{Q} \right)/\left( 1 - x_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | d{f} = \left( y_{Q} - y^{2}_{Q} \right) / \left( x_{Q} - x^{2}_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \boldsymbol{x_{1}=x_{Q}/3 }
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | x_{2}= x_{Q} \left( 1 + x_{Q} - x_{Q} \right) / y_{Q}
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | === 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)
| + | |
- | | + | |
- | [[Image: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
| + | |
- | | + | |
- | Based on Fromm's scheme
| + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_Schemes_MUSCL_Probe_01.jpg]]
| + | |
- | | + | |
- | | + | |
- | Normalized variables - uniform grids
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | \hat{\phi_{f}}=
| + | |
- | \begin{cases}
| + | |
- | 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{4} \\
| + | |
- | \frac{1}{4} + \hat{\phi_{C}} & \frac{1}{4} \leq \hat{\phi_{C}} \leq \frac{3}{4} \\
| + | |
- | 1 & \frac{3}{4} \leq \hat{\phi_{C}} \leq 1 \\
| + | |
- | \widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\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}
| + | |
- | 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q}/2 \\
| + | |
- | a_{f} + b_{f} \hat{\phi_{C}} & x_{Q}/2 \leq \hat{\phi_{C}} \leq 3 x_{Q}/2 \\
| + | |
- | 1 & 3 x_{Q}/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>
| + | |
- | | + | |
- | where
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | a_{f}= \left( 3 x_{Q} - 2 \right)/2
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | b_{f}= \left( 1 - x_{Q} \right) / x_{Q}
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | === 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
| + | |
- | | + | |
- | == Schemes by Leonard ==
| + | |
- | === 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,
| + | |
- | 5:341–377, 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,
| + | |
- | 124:2588–2606, 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 ==
| + | |
- | | + | |
- | == Sweby <math>\Phi</math> - 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
| + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]]
| + | |
- | | + | |
- | == SOUCUP - Second-Order Upwind Central differnce-first order UPwind ==
| + | |
- | | + | |
- | {{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}} | + | |
- | | + | |
- | {{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 }}
| + | |
- | | + | |
- | | + | |
- | [[Image:NM_convectionschemes_struct_grids_Schemes_SOUCUP_Probe_01.jpg]]
| + | |
- | | + | |
- | Normalized variables - uniform grids
| + | |
- | | + | |
- | <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
| + | |
- | \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}} & 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}
| + | |
- | </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>
| + | |
- | \boldsymbol{b_{f}= y_{Q}/x_{Q} }
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | c_{f}= \left( x_{Q} - y_{Q} \right)/\left( 1 - x_{Q} \right)
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<math>
| + | |
- | 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 ==
| + | |
- | | + | |
- | 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'''
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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 ==
| + | |
- | | + | |
- | '''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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- | | + | |
- | | + | |
- | -----------------------------------------------------------------
| + | |
- | | + | |
- | | + | |
- | 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
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- | | + | |
- | <table width="100%"><tr><td>
| + | |
- | :<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
| + | |
- | \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>
| + | |
- | \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 \\
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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>
| + | |
- | </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
| + | |
- | \end{cases}
| + | |
- | </math>
| + | |
- | </td><td width="5%">(2)</td></tr></table>
| + | |
- | | + | |
- | where
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- | | + | |
- | <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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- | | + | |
- | | + | |
- | 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>
| + | |
- | | + | |
- | --------------------------------------------------------
| + | |
- | 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>
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- | | + | |
- | for <math>\boldsymbol{U_w \triangleleft 0}</math>
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- | | + | |
- | <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>
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- | | + | |
- | [[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
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- | | + | |
- | | + | |
- | == 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 ==
| + | |
- | | + | |
- | == GAMMA ==
| + | |
| | | |
- | == CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection ==
| + | ---- |
| + | <i> Return to [[Numerical methods | Numerical Methods]] </i> |
| | | |
- | '''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 [[Approximation Schemes for convective term - structured grids]] </i> |