Page 84 - Introduction to Computational Fluid Dynamics

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3.8 TOTAL VARIATION DIMINISHING SCHEME
considerations associated with false diffusion in multidimensional flows will be 10:54 63
discussed in Chapter 5.
3.7 Hybrid and Power-Law Schemes
Spalding [75] derived a hybrid difference scheme (HDS) such that, in Equation 3.20,
ψ is given by
1 P c
ψ = P c − 1 + max −P c , 1 − , 0 (HDS). (3.36)
2
P c
Similarly, Patankar [49] argued that the best representation is the exact solution
itself (see Equation 3.28). However, this will require evaluation of exponential terms
and this is not economically attractive in practical computing. Therefore, he chose
to mimic Equation 3.28 through a power-law scheme, which implies that

ψ = [P c − 1 + max(0, −P c )] /P c
5
+ max 0, (1 − 0.1|P c |) /P c (Power law). (3.37)
With these two expressions for ψ, it is now possible to construct AE and AW
coefficients (see Equation 3.25) for the HDS and power-law schemes. The resulting
implications for P are tabulated in Table 3.1. Notice that for |P c |≤ 2, the HDS
results match exactly with those of the CDS. For |P c | > 2, the HDS assumes that
|P c |=∞ or, in other words, conduction flux is set to zero. This may be considered
too drastic but it nonetheless ensures positivity of coefficients for all values of P c .
The results from the power-law scheme, of course, do mimic the exact solution
quite well.

3.8 Total Variation Diminishing Scheme

The difference schemes discussed so far are found to be adequate when the spatial
variation of is expected to be smooth and continuous. Often, however, the
variation is almost discontinuous (as across a shock). To capture such variation,
extremely small values of x become necessary, resulting in uneconomic com-
putations. However, if coarse grids are employed then UDS, HDS, or power-law
schemes produce smeared shock predictions.
Total variation diminishing (TVD) schemes enable sharper shock predictions
on coarse grids. In these schemes, in addition to magnitude and sign of P c , the
nature of the variation of in the neighbourhood of node P is also sensed. Thus,
instead of Equations 3.21 and 3.22, we write
1
+
+
P e = (P +|P|) f E + (1 − f ) W
e
e
2
1
−
−
+ (P −|P|) f P + (1 − f ) EE , (3.38)
e
e
2