Approximation Schemes for convective term - structured grids - Schemes
From CFD-Wiki
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- | + | == Linear == | |
- | + | === 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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AIAA J. 14 (1976) 1241-1249 | 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) | '''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 | '''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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</td><td width="5%">(2)</td></tr></table> | </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) | '''H.C.Price, R.S. Varga and J.E.Warren''' , Application of oscillation matrices to diffusion-convection equations, Journal Math. and Phys., Vol. 45, p.301, (1966) | ||
- | + | === Fromm - Fromm's Upwind Scheme === | |
- | + | === CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4) === | |
In CUDS (UDS-4) for interpolation of function is used three upwind nodes and one node downstream. | In CUDS (UDS-4) for interpolation of function is used three upwind nodes and one node downstream. | ||
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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. | ||
- | + | === CUI - Cubic Upwind Interpolation === | |
B.P. Leonard | B.P. Leonard | ||
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--------------------------------------------------------------------- | --------------------------------------------------------------------- | ||
- | + | == Non-Linear QUICK based == | |
- | + | === SMART - Sharp and Monotonic Algorithm for Realistic Transport === | |
'''P.H.Gaskell and A.C.K. Lau''', Curvature-compensated convective transport: SMART, a new boundedness preserving transport algorithm, International J. Numer. Methods Fluids 8 (1988) 617-641 | '''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 | ||
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[[Image:NM_convectionschemes_struct_grids_SMART_probe_01.jpg]] | [[Image:NM_convectionschemes_struct_grids_SMART_probe_01.jpg]] | ||
- | + | === SMARTER - SMART Efficiently Revised === | |
J.K. Shin and Y.D. Choi | J.K. Shin and Y.D. Choi | ||
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Trans. KSME 16(6) (1992) 1179-1194 (in Korean) | Trans. KSME 16(6) (1992) 1179-1194 (in Korean) | ||
- | + | === WACEB === | |
Song B., Liu G.B., Kam K.Y., Amano R.S. | Song B., Liu G.B., Kam K.Y., Amano R.S. | ||
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International Journal for Numerical Methods in Fluids, 2000, 32, 881-897 | International Journal for Numerical Methods in Fluids, 2000, 32, 881-897 | ||
- | + | === VONOS - Variable-Order Non-Oscillatory Scheme === | |
'''Varonos A., Bergeles G.''', Development and assessment of a Variable-Order Non-oscillatory Scheme for convection term discretization // International Journal for Numerical Methods in Fluids. 1998. 26, N 1. 1-16 | '''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) | '''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) | ||
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Lecture notes in Physics, No. 453, pp. 372-377, Springler-Verlag, Berlin, (1995) | 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) | '''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) | ||
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--------------------------------------------------------------------- | --------------------------------------------------------------------- | ||
- | + | == Fromm based == | |
- | + | === Fromm scheme === | |
J.E.Fromm | J.E.Fromm | ||
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J. Comp. Phys., Vol. 3, p.176, (1968) | 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 | '''Lien F.S. and Leschziner M.A.''' , Proc. 5th Int. IAHR Symp. on Refind Flow Modelling and Turbulence Measurements, Paris, Sept. 1993 | ||
- | + | === van Leer limiter === | |
- | + | === van Albada === | |
Bounded Fromm | Bounded Fromm | ||
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Astron. Astrophysics, Vol. 108, p.76, 1982 | Astron. Astrophysics, Vol. 108, p.76, 1982 | ||
- | + | === OSPRE === | |
bounded Fromm | bounded Fromm | ||
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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 | In C. Taylor and P.Durbetaki, editors, Proc. Ninth Int. Conf. on Numer. Method. Laminar and turbulent Flow, pages 203-214, Pineride Press, Swansea, 1995 | ||
- | + | == ULTIMATE Universal Limiter == | |
- | + | == Chakravarthy-Osher limiter == | |
- | + | == Sweby \Phi - limiter == | |
- | + | == Superbee limiter == | |
- | + | == R-k limiter == | |
- | + | == MINMOD - MINimum MODulus == | |
'''Harten A.''' High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393 | '''Harten A.''' High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393 | ||
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[[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]] | [[Image:NM_convectionschemes_struct_grids_MINMOD_probe_01.jpg]] | ||
- | + | == ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars == | |
Third-order flux-limiter scheme | Third-order flux-limiter scheme | ||
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- | + | == 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=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 }} | {{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 }} | ||
- | + | == COPLA - COmbination of Piecewise Linear Approximation == | |
'''Seok Ki Choi, Ho Yun Nam, Mann Cho''' | '''Seok Ki Choi, Ho Yun Nam, Mann Cho''' | ||
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Numerical Heat Transfer, Part B, 28:23-38, 1995 | Numerical Heat Transfer, Part B, 28:23-38, 1995 | ||
- | + | == HLPA - Hybrid Linear / Parabolic Approximation == | |
'''Zhu J'''. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232. | '''Zhu J'''. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232. | ||
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</td><td width="5%">(2)</td></tr></table> | </td><td width="5%">(2)</td></tr></table> | ||
- | + | == 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 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. | G. Papadakis, G. Bergeles. | ||
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Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995 | Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995 | ||
- | + | == Koren == | |
bounded CUS | bounded CUS | ||
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In: Numerical Mthods for Advection-Diffusion Problems, Ed. C.B.Vreugdenhil& B.Koren, Vieweg, Braunscheweigh, p.117, (1993) | In: Numerical Mthods for Advection-Diffusion Problems, Ed. C.B.Vreugdenhil& B.Koren, Vieweg, Braunscheweigh, p.117, (1993) | ||
- | + | == H-CUS == | |
bounded CUS | bounded CUS | ||
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VKI-preprint, 1995-21, (1995) | VKI-preprint, 1995-21, (1995) | ||
- | + | == MLU == | |
B. Noll | B. Noll | ||
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AIAA J., vol. 30, No. 1, p.64 (1992) | 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 | '''B.P.Leonard''', Simple high-accuracy resolution rogram for convective modelling of discontinuities, International J. Numerical Methods Fluids 8 (1988) 1291-1381 | ||
- | + | == LPPA - Linear and Piecewise / Parabolic Approximasion == | |
- | + | == GAMMA == | |
- | + | == CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection == | |
'''M.A. Alves, P.J.Oliveira, F.T. Pinho''', A convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection // International Lournal For Numerical Methods in Fluids 2003, 41; 47-75 | '''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 |
Revision as of 03:41, 18 September 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
Linear
SOU - Second Order Upwind (also LUDS or UDS-2)
S.P.Vanka ({{{year}}}), "Second-order upwind differencing ina recirculating flow", AIAA J., 25, 1435-1441.
R.F.Warming and R.M. Beam
Upwind second order difference schemes and applications in aerodynamics flows
AIAA J. 14 (1976) 1241-1249
Skew - Upwind
G.D.Raithby , Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)
QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS)
B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98
Usual variables
| (2) |
Normalised variables (uniform grid)
| (2) |
Normalised variables (non-uniform grid)
| (2) |
LUS - Linear Upwind Scheme
H.C.Price, R.S. Varga and J.E.Warren , Application of oscillation matrices to diffusion-convection equations, Journal Math. and Phys., Vol. 45, p.301, (1966)
Fromm - Fromm's Upwind Scheme
CUDS - Cubic Upwind Difference Scheme (also CUS or UDS-4)
In CUDS (UDS-4) for interpolation of function is used three upwind nodes and one node downstream.
usual variables
| (2) |
normalised variables (uniform grids)
| (2) |
R.K. Aragval
A third-order-accurate upwind scheme for Navier-Stokes solution at high Reynolds numbers
Paper No. AIAA-81-0112, AIAA 19th Aerospace Science Meeting, St. Louis, 1982.
CUI - Cubic Upwind Interpolation
B.P. Leonard
A survey of finite differences of opinion on numerical muddling of incompressible defective confusion equation
paper in ASME, Applied Mechanics Division, Winter Annual Meeting, 1979
Non-Linear QUICK based
SMART - Sharp and Monotonic Algorithm for Realistic Transport
P.H.Gaskell and A.C.K. Lau, Curvature-compensated convective transport: SMART, a new boundedness preserving transport algorithm, International J. Numer. Methods Fluids 8 (1988) 617-641
SMARTER - SMART Efficiently Revised
J.K. Shin and Y.D. Choi
Study on the improvement of the convective differencing scheme for the high-accuracy and stable resolution of the numerical solution
Trans. KSME 16(6) (1992) 1179-1194 (in Korean)
WACEB
Song B., Liu G.B., Kam K.Y., Amano R.S.
On a higher-order bounded discretization schemes
International Journal for Numerical Methods in Fluids, 2000, 32, 881-897
VONOS - Variable-Order Non-Oscillatory Scheme
Varonos A., Bergeles G., Development and assessment of a Variable-Order Non-oscillatory Scheme for convection term discretization // International Journal for Numerical Methods in Fluids. 1998. 26, N 1. 1-16
CHARM - Cubic / Parabolic High-Accuracy Resolution Method
G.Zhou , Numerical simulations of physical discontinuities in single and multi-fluid flows for arbitrary Mach numbers, PhD Thesis, Chalmers University of Technology, Sweden (1995)
Gang Zhou, Lars Davidson and Erik Olsson
Transonic Inviscid / Turbulent Airfoil Flow Simulations Using a Pressure Based Method with High Order Schemes
Lecture notes in Physics, No. 453, pp. 372-377, Springler-Verlag, Berlin, (1995)
UMIST - Upstream Monotonic Interpolation for Scalar Transport
F.S.Lien and M.A.Leschziner , Upstream Monotonic Interpolation for Scalar Transport with application to complex turbulent flows, International Journal for Numerical Methods in Fluids, Vol. 19, p.257, (1994)
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
van Leer limiter
van Albada
Bounded Fromm
G.D. Van Albada, B.Van Leer, W.W.Roberts
A comparative study of computational methods in cosmic gas dynamics
Astron. Astrophysics, Vol. 108, p.76, 1982
OSPRE
bounded Fromm
Waterson [1995]
N.P.Waterson, H.Deconinck.
A unified approach to the design and application of bounded high-order convection schemes
In C. Taylor and P.Durbetaki, editors, Proc. Ninth Int. Conf. on Numer. Method. Laminar and turbulent Flow, pages 203-214, Pineride Press, Swansea, 1995
ULTIMATE Universal Limiter
Chakravarthy-Osher limiter
Sweby \Phi - limiter
Superbee limiter
R-k limiter
MINMOD - MINimum MODulus
Harten A. High resolution schemes using flux limiters for hyperbolic conservation laws. Journal of Computational Physics 1983; 49: 357-393
A. Harten
High Resolution Schemes for Hyperbolic Conservation Laws
J. Comp. Phys., vol. 49, no. 3, pp. 225-232, 1991
ISNAS - Interpolation Scheme which is Nonoscillatory for Advected Scalars
Third-order flux-limiter scheme
M. Zijlema , On the construction of a third-order accurate monotone convection scheme with application to turbulent flows in general domains. International Journal for numerical methods in fluids, 22:619-641, 1996.
SOUCUP - Second-Order Upwind Central differnce-first order UPwind
Zhu J. (1992), "On the higher-order bounded discretization schemes for finite volume computations of incompressible flows", Computational Methods in Applied Mechanics and Engineering. 98. 345-360.
J. Zhu, W.Rodi (1991), "A low dispersion and bounded convection scheme", Comp. Meth. Appl. Mech.&Engng, Vol. 92, p 225.
COPLA - COmbination of Piecewise Linear Approximation
Seok Ki Choi, Ho Yun Nam, Mann Cho
Evaluation of a High-Order Bounded Convection Scheme: Three-Dimensional Numerical Experiments
Numerical Heat Transfer, Part B, 28:23-38, 1995
HLPA - Hybrid Linear / Parabolic Approximation
Zhu J. Low Diffusive and oscillation-free convection scheme // Communications and Applied Numerical Methods. 1991. 7, N3. 225-232.
Zhu J., Rodi W. A low dispersion and bounded discretization schemes for finite volume computations of incompressible flows // Computational Methods for Applied Mechanics and Engineering. 1991. 92. 87-96
In this scheme, the normalized face value is approximated by a combination of linear and parabolic charachteristics passing through the points, O, Q, and P in the NVD. It satisfies TVD condition and is second-order accurate
Usual variables
| (2) |
Normalized variables - uniform grids
| (2) |
Normalized variables - non-uniform grids
| (2) |
where
| (2) |
Implementation
Using the switch factors:
for
| (2) |
for
| (2) |
and taken all the possible flow directions into account, the un-normalized form of equation can be written as
| (2) |
where
| (2) |
| (2) |
CLAM - Curved-Line Advection Method
Van Leer B. , Towards the ultimate conservative difference scheme. II. Monotonicity and conservation combined in a second-order scheme. Journal of Computational Physics 1974; 14:361-370
van Leer harmonic
BSOU
G. Papadakis, G. Bergeles.
A locally modified second order upwind scheme for convection terms discretization.
Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995
Koren
bounded CUS
B. Koren
A robust upwind discretisation method for advection, diffusion and source terms
In: Numerical Mthods for Advection-Diffusion Problems, Ed. C.B.Vreugdenhil& B.Koren, Vieweg, Braunscheweigh, p.117, (1993)
H-CUS
bounded CUS
N.P.Waterson H.Deconinck
A unified approach to the design and application of bounded high-order convection schemes
VKI-preprint, 1995-21, (1995)
MLU
B. Noll
Evaluation of a bounded high-resolution scheme for combustor flow computations
AIAA J., vol. 30, No. 1, p.64 (1992)
SHARP - Simple High Accuracy Resolution Program
B.P.Leonard, Simple high-accuracy resolution rogram for convective modelling of discontinuities, International J. Numerical Methods Fluids 8 (1988) 1291-1381
LPPA - Linear and Piecewise / Parabolic Approximasion
GAMMA
CUBISTA - Convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection
M.A. Alves, P.J.Oliveira, F.T. Pinho, A convergent and Universally Bounded Interpolation Scheme for the Treatment of Advection // International Lournal For Numerical Methods in Fluids 2003, 41; 47-75