Generalisation of upwind-biased discretisation schemes Consider the standard control volume discretisation of the one-dimensional

5.10.1 Generalisation of upwind-biased discretisation schemes Consider the standard control volume discretisation of the one-dimensional

convection–diffusion equation (5.3). Discretisation of the diffusion terms using the central differencing practice is standard and does not require any further consideration. It is the discretisation of the convective flux term that requires special attention. We assume that the flow is in the positive x -direction, so u > 0, and develop the TVD concept as a generalisation of

upwind-biased expressions for the value of transported quantity φ at the east face of a one-dimensional control volume.

The standard upwind differencing (UD) scheme for the east face value of φ e gives

φ e = φ P (5.64)

A linear upwind differencing (LUD) scheme, which involves two upstream

values, yields the following expression for φ e :

( φ P − φ W ) δx φ e = φ P + δx

1 = φ P +( φ P − φ W )

166 CHAPTER 5 FINITE VOLUME METHOD FOR C---D PROBLEMS

This can be thought of as a second-order extension of the original UD estimate (5.64) of φ e with a correction based on an upwind-biased estimate ( φ P − φ W )/ δx of the gradient of φ multiplied by the distance δx/2 between node P and the east face. Another way of looking at this is to recall that our

aim is to construct expressions for convective flux F e φ e . Hence, for positive flow direction, the convective flux discretisation by means of the LUD scheme can be thought of as the sum of the basic UD convective flux F e φ P plus an additional flux contribution F e ( φ P − φ W )/2 to improve the order of accuracy. The QUICK scheme (5.47) can be similarly rearranged in the form of the UD estimate plus a correction:

φ e = φ P + [3 φ E −2 φ P − φ W ]

The central differencing (CD) scheme can be written as follows:

= φ P +( φ E − φ P )

We consider a generalization of the higher-order schemes in the following form:

φ e = φ P + ψ(φ E − φ P )

2 where ψ is an appropriate function.

In choosing this form we express the convective flux at the east face as the sum of the flux F e φ P that is obtained when we use UD and an additional con- vective flux F e ψ(φ E − φ P )/2. The extra contribution is connected in some way to the gradient of the transported quantity φ at the east face, as indicated by its central difference approximation ( φ E − φ P ). It is easy to see that the central difference scheme (5.68) leads to function ψ = 1, but in sections 5.3–5.5 we have established that an additional convective flux based on this choice of ψ leads to wiggles in the solution if the grid is too coarse due to lack of transportiveness. The upwind scheme (5.64) corresponds to function ψ = 0, but this choice of ψ gave rise to false diffusion. Looking at the higher- order schemes we find that the LUD scheme (5.65) may be rewritten as

Hence, for LUD the function is ψ = (φ P − φ W )/( φ E − φ P ). After some algebra the QUICK expression (5.66) can be rewritten as

By comparing equation (5.70) with equation (5.68) it can be seen that the appropriate function for the QUICK scheme is

ψ=3+ B

5.10 TVD SCHEMES

Inspection of the forms (5.69) and (5.70) shows that the ratio of upwind-side gradient to downwind-side gradient ( φ P − φ W )/( φ E − φ P ) determines the value of function ψ and the nature of the scheme. Therefore, we let

The general form of the east face value φ e within a discretisation scheme for

convective flux may be written as

φ e = φ P + ψ(r)(φ E − φ P )

2 For the UD scheme ψ(r) = 0

For the CD scheme ψ(r) = 1 For the LUD scheme ψ(r) = r For the QUICK scheme ψ(r) = (3 + r)/4

Figure 5.21 shows the ψ(r) vs. r relationships for these four schemes. This diagram is known as the r – ψ diagram. All the above expressions assume that the flow direction is positive (i.e. from west to east). It can be shown that similar expressions exist for negative flow direction and r will still be the ratio

of upwind-side gradient to downwind-side gradient.

Figure 5.21 The function ψ for various discretisation schemes