Approximation Schemes for convective term - structured grids - Schemes

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Revision as of 20:17, 14 October 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

reference shablon

SMITH (3000), "XXX", XXX.

fds

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

 $\hat{\phi_{f}}= \begin{cases} 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{4} \\ \frac{1}{4} + \hat{\phi_{C}} & \frac{1}{4} \leq \hat{\phi_{C}} \leq \frac{3}{4} \\ 1 & \frac{3}{4} \leq \hat{\phi_{C}} \leq 1 \\ \widehat{\phi_{C}} & \widehat{\phi_{C}} \triangleleft 0 \ , \ \widehat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} 2 \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q}/2 \\ a_{f} + b_{f} \hat{\phi_{C}} & x_{Q}/2 \leq \hat{\phi_{C}} \leq 3 x_{Q}/2 \\ 1 & 3 x_{Q}/2 \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $a_{f}= \left( 3 x_{Q} - 2 \right)/2$ (2)
 $b_{f}= \left( 1 - x_{Q} \right) / x_{Q}$ (2)

van Leer limiter

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.

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.

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

 $\hat{\phi_{f}}= \begin{cases} \frac{3}{2} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq \frac{1}{2} \\ \frac{1}{2} + \frac{1}{2} \hat{\phi_{C}} & \frac{1}{2} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f}+ b_{f} \hat{\phi_{C}} & 0 \leq \hat{\phi_{C}} \leq x_{Q} \\ c_{f}+ d_{f} \hat{\phi_{C}} & x_{Q} \leq \hat{\phi_{C}}\leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $\boldsymbol{b_{f}= y_{Q}/x_{Q} }$ (2)
 $c_{f}= \left( x_{Q} - y_{Q} \right)/\left( 1 - x_{Q} \right)$ (2)
 $d_{f} = \left( 1 - y_{Q} \right) / \left( 1 - x_{Q} \right)$ (2)

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

 $f_{w}= \begin{cases} f_{w} + \left( f_{P} - f_{W} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\ f_{W} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - uniform grids

 $\hat{f_{w}}= \begin{cases} \hat{f_{C}} \left( 2 - \hat{f_{C}} \right) \hat{f_{C}} & 0 \leq \hat{f_{C}} \leq 1 \\ \hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{f_{w}}= \begin{cases} a_{w} + b_{w} \hat{f_{C}} + c_{w} \hat{f_{C}}^{2} & 0 \leq \hat{f_{C}} \leq 1 \\ \hat{f_{C}} & \hat{f_{C}} \triangleleft 0 \ , \ \hat{f_{C}} \triangleright 1 \end{cases}$ (2)

where

 $a_{w} = 0 , b_{w} = \left(y_{Q}- x^{2}_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) , c_{w} = \left(y_{Q}- x_{Q} \right) / \left(x_{Q}- x^{2}_{Q} \right) ,$ (2)

Implementation

Using the switch factors:

for $\boldsymbol{U_w \geq 0}$

 $\alpha^{+}_{w} = \begin{cases} 1 & \ if \ | \phi_{P} - 2 \phi_{W} + \phi_{WW}| \triangleleft | \phi_{P} - \phi_{WW} | \\ 0 & otherwise \end{cases}$ (2)

for $\boldsymbol{U_w \triangleleft 0}$

 $\alpha^{-}_{w} = \begin{cases} 1 & \ if \ | \phi_{W} - 2 \phi_{P} + \phi_{E}| \triangleleft | \phi_{W} - \phi_{E} | \\ 0 & otherwise \end{cases}$ (2)

and taken all the possible flow directions into account, the un-normalized form of equation can be written as

 $\phi_{w} = U^{+}_{w} \phi_{W} + U^{-}_{w} \phi_{P} + \Delta \phi_{w}$ (2)

where

 $\Delta \phi_{w} = U^{+}_{w} \alpha^{+}_{w} \left( \phi_{P} - \phi_{W} \right) \frac{\phi_{W} - \phi_{WW}}{\phi_{P} - \phi_{WW}} + U^{-}_{w} \alpha^{-}_{w} \left( \phi_{W} - \phi_{P} \right) \frac{\phi_{P} - \phi_{E}}{\phi_{W} - \phi_{E}}$ (2)
 $U^{+}_{w} = 0.5 \left( 1 + \left| U_{w} \right| / U_{w} \right) \ , \ U^{-}_{w} = 1 - U^{+}_{w} \ \ \left( U_{w}\neq 0 \right)$ (2)

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

BSOU

A locally modified second order upwind scheme for convection terms discretization.

Int. J. Numer. Meth. Heat Fluid Flow, 5.49-62, 1995

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

 $\hat{\phi_{f}}= \begin{cases} \frac{9}{4}{\phi}_{C} - \frac{3}{2} {\phi}^{2}_{C} & 0 \leq \hat{\phi}_{C} \leq \frac{1}{2} \\ \frac{1}{4}+\frac{5}{4}{\phi}_{C}-\frac{1}{2}{\phi}^{2}_{C} & \frac{1}{2} \leq \hat{\phi}_{C} \leq 1 \\ \hat{\phi}_{C} & \hat{\phi}_{C} \triangleleft 0 \ , \ \hat{\phi}_{C} \triangleright 1 \end{cases}$ (2)

Normalized variables - non-uniform grids

 $\hat{\phi_{f}}= \begin{cases} a_{f}+ b_{f} \hat{\phi}_{C} + c_{f}\hat{\phi}^{2}_{C} & 0 \leq \hat{\phi_{C}} \leq x_{Q} \\ d_{f}+ d_{f} \hat{\phi}_{C} + f_{f}\hat{\phi}^{2}_{C} & x_{Q} \leq \hat{\phi_{C}} \leq 1 \\ \hat{\phi_{C}} & \hat{\phi_{C}} \triangleleft 0 \ , \ \hat{\phi_{C}} \triangleright 1 \end{cases}$ (2)

where

 $\boldsymbol{a_{f}= 0}$ (2)
 $b_{f}= \left( s_{Q}x^{2}_{Q} + 2x_{Q}y_{Q} \right) / x^{2}_{Q}$ (2)
 $c_{f}= \left( s_{Q}x_{Q} - y_{Q} \right)/ x^{2}_{Q}$ (2)
 $d_{f} = \left[ x^{2}_{Q} + s_{Q} \left( x^{2}_{Q} - x_{Q} \right)+ \left( 1 - 2 x_{Q} \right) \right] / \left( 1 - x_{Q} \right)^{2}$ (2)
 $e_{f} = \left[ -2 x_{Q} + s_{Q} \left( 1 - x^{2}_{Q} \right) + 2 x_{Q} y_{Q} \right] / \left( 1 - x_{Q} \right)^{2}$ (2)
 $f_{f} = \left[ 1 + s_{Q} \left( x_{Q} - 1 \right) - 2 y_{Q} \right] / \left( 1 - x_{Q} \right)^{2}$ (2)

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