# Linear Schemes - structured grids

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## Introduction

Linear schemes UDS, CDS and these

## 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

 $\phi_{w}= \frac{3}{8}\phi_{P}+ \frac{3}{4}\phi_{W} - \frac{1}{8}\phi_{WW}$ (2)
 $\phi_{f}= \frac{3}{8}\phi_{D}+ \frac{3}{4}\phi_{C} - \frac{1}{8}\phi_{U}$ (2)

Normalised variables (uniform grid)

 $\hat{\phi_{f}}= \frac{3}{8} + \frac{3}{4}\hat{\phi_{C}}$ (2)

Normalised variables (non-uniform grid)

 $\begin{matrix} \hat{\phi_{w}} & = \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{W}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right] \right\} U^{+}_{w} + \\ + & \left\{ C_{2} \left( 1 + C_{3} \right) \hat{\phi_{P}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right] \right\} U^{-}_{w} \end{matrix}$ (2)
 $\begin{matrix} \hat{\phi_{f}} & = \left\{ \left( 1 + C_{1} \right) \left( 1 - C_{2} \right)\hat{\phi_{C}} + C_{2} \left[ 1 - \frac{C_{1} \left( 1 - C_{2} \right) }{ C_{1} + C_{2} } \right] \right\} U^{+}_{f} + \\ + & \left\{ C_{2} \left( 1 + C_{3} \right) \hat{\phi_{D}} + \left( 1 - C_{2} \right) \left[ 1 - \frac{C_{2} C_{3} }{ 1- C_{2} + C_{3} } \right] \right\} U^{-}_{f} \end{matrix}$ (2)

## LUS - Linear Upwind Scheme

H.C.Price, R.S. Varga and J.E.Warren (1966), "Application of oscillation matrices to diffusion-convection equations", Journal Math. and Phys., Vol. 45, p.301, (1966).

 $\phi_{f}= \phi_{C} + 0.5 \left( \phi_{C} - \phi_{U} \right)$ (1)

## 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

 $\phi_{w}=\frac{1}{3}\phi_{P} + \frac{5}{6}\phi_{W} + \frac{1}{6}\phi_{WW}$ (2)

normalised variables (uniform grids)

 $\hat{\phi_{w}}=\frac{1}{3} + \frac{5}{6}\hat{\phi_{W}}$ (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