Approximation Schemes for convective term - structured grids - Schemes
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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
reference shablon
SMITH (3000), "XXX", XXX.
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 (1976), "Upwind second order difference schemes and applications in aerodynamics flows", AIAA J. 14 (1976) 1241-1249.
Skew - Upwind
G.D.Raithby , Skew upstream differencing schemes for problems involving fluid flow, Computational Methods Applied Mech. Engineering, 9, 153-164 (1976)
QUICK - Quadratic Upwind Interpolation for Convective Kinematics (also UDS-3 or QUDS)
B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98
Usual variables
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Normalised variables (uniform grid)
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Normalised variables (non-uniform grid)
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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
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normalised variables (uniform grids)
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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
Normalized variables - uniform grids
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Normalized variables - non-uniform grids
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where
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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
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Normalized variables - non-uniform grids
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where
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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
Normalized variables - uniform grids
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Normalized variables - non-uniform grids
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where
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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
Based on Fromm's scheme
Normalized variables - uniform grids
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Normalized variables - non-uniform grids
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where
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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 - 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
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.
Normalized variables - uniform grids
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Normalized variables - non-uniform grids
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where
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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
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
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Normalized variables - uniform grids
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Normalized variables - non-uniform grids
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where
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Implementation
Using the switch factors:
for
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for
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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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where
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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
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