Approximation Schemes for convective term - structured grids - Schemes
From CFD-Wiki
Schemes
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)
B.P.Leonard, A stable and accurate modelling procedure based on quadratic interpolation, Comput. Methods Appl. Mech. Engrg. 19 (1979) 58-98
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)
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
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
OSPRE
ULTIMATE Universal Limiter
Chakravarthy-Osher limiter
Sweby \Phi - limiter
Superbee
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
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
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
Summary of Discretizations Schemes
Discretizations Schemes Estimation of order
Selection advice
Comparison of Discretizations Schemes
Numerical examples
Pure convection of a scalar step by a rotating velocity field (Smith-Hutton test)
R.M.Smith and A.G.Hutton (1982), "The numerical treatment of advection: A performance comparison of current methods", Numerical Heat Transfer, Vol. 5, p439.