1
“Variable parameter Muskingum-Cunge method for flood routing
in a compound channel”
by
Tang X, Knight D.W. and Samuels P.G.
J.Hyd.Res. vol 37, 1999, No. 5.
Discussion by Alan A. Smith
The paper describes the use of a non-centered finite difference scheme that transforms the
kinematic equation into a diffusion equation. The routing solution exhibits a numerical
error that results in attenuation and lag of the outflow hydrograph. The trick is then to
select appropriate values for the weighting coefficients in time and space in order to
cause the numerical diffusion to match the physical phenomenon. This discussion
concerns the conditions required for numerical stability and describes a method that
automatically ensures selection of the correct increments in time and space for any
desired values of the weighting coefficients.
The authors’ Fig. 1 shows an element of the computational grid in which the spatial
weighting factor  and the temporal weighting factor  are defined. However, following
eq. (4) the assumption is made that = 0.5 so that the finite difference terms are assumed
to be centered in time throughout the remainder of the paper.
This discussion suggests that there is some value in maintaining some generality with
both temporal and spatial weighting factors. A more general depiction of the space-time
module is shown in Figure 1 below,
Figure 1 – Stability and numerical error characteristics of a space-time element.
The weighting coefficients  and  are those used in Smith (1980) and correspond to the
authors’  and  respectively with the difference that  The size of the space-time
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element is x wide and t high. The diagonal with slope -c defines the maximum
time-step tc which is required to satisfy the Courant condition for numerical stability.
Expressing the continuity equation as:
1
c
A A

0
x t
assuming zero lateral inflow q
Biesenthal (1974) applied the Von Neumann method for analyzing errors and obtained
the following condition for numerical stability.
2
  1   Cr      Cr  1     1   Cr  1      Cr
 0
where Cr is the Courant stability number given by [3].
3
Cr 
t
t c
 c
t
x
The absolute functions can be removed from the first and last terms of eq. [2] since
4
0.0   , 
 1.0
so that the criterion [2] becomes
5
 1  2   1  2  Cr      Cr  1     1   Cr  0
Examination of the four possible solutions to inequality [5] leads to the stability criterion
6
1   

 Cr 
1   

Figure 1 shows a case in which Cr is approximately equal to 0.5 and from [6] it follows
that centering the finite differences in the region below and to the left of the main
diagonal would result in an unstable solution. It is demonstrated later in this discussion
that eq. [6] corresponds to the authors’ equation (37).
Now for values of  and  that satisfy eq. [6] the error resulting from the imperfect
centering can be expressed as follows.
[7 ]
Q A
x  2 Q
t  2 A



 2  1

1

2

x t
2 x 2
2 t 2
x 2  3Q
 1  3  3 2
 1  3  3 2
3
6 x
2
t 2  3Q
2 x
  2




2 xt 2
2







 6t
3 A
t 3
3 A

x 2 t
2
0
Two conclusions emerge from [7].
(a)
As the increments x and t reduce to zero the expression converges to the
differential equation and is independent of the values of  and 
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(b)
As  and  depart from the central position of (0.5,0.5), truncation errors of the
order O(x) and O(t) increase respectively and independently.
Including only the first order terms in [7] it can be shown that the diffusion coefficient
can be expressed as:
[8]
x
t 

D   2  1  1  2  c 
2
2

x
1  2   2   1Cr 

2
By setting  = 0.5 in eq. [8] and substituting for  in the inequality [6] one obtains the
more specific criteria of the authors’ eq. (37).
This leads to the conclusion that a line parallel to the leading diagonal in Figure 1
represents the locus of a set of (,) coordinate pairs which will produce the same value
of numerical error. Also, the error increases linearly as the lines approach the upper right
corner of the space-time element defined by =0.0, =1.0.
As mentioned by the authors, neglect of the convective and temporal acceleration terms
in the St Venant equation allows the diffusion coefficient to be expressed in terms of
channel characteristics. Thus:
[9]
D
Q
2S dQ

dh
Q
2 BSc
(see authors eq. (3))
Combining eqs. [8] and [9] provides a method for evaluating the weighting coefficients 
and  to be evaluated for given channel characteristics. Thus
[10]
1  2   2  1Cr 
Q
h f dQ
where
h f  S x
dh
A suitable algorithm to implement this scheme is summarized as follows.
(a) Determine the diffusion coefficient from [9].
(b) Assume =0.5 and calculate maximum reach length xmax (from 1st & 2nd terms of
[6]) and tmax (from 2nd & 3rd terms of [6]).
(c) Subdivide x or t of space-time element into sub-multiples to ensure stability. For
very long channels use a cascade of N channel segments each of length L/N. For
very short channels use time-steps t = t/N.
(d) Calculate  from [8].
(e) If  < 0.0 then set  = 0.0 and compute  in the range 0.5 ≤ ≤ 1.0 from [8].
(f) Compute the Muskingum-Cunge coefficients in terms of ,  and Cr, i.e.
Note: C4  C1  C2  C3
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[11]
C1    1   Cr
C 2  1     1   Cr
C 3     Cr
C 4  1      Cr
where Cr  c
t
x
(g) Route the inflow through single or multiple reaches of the system using interpolation
of the inflow in cases where t < t.
The above algorithm has been implemented in the Miduss program since the initial DOS
version over 10 years ago, however the channel cross-section was initially limited to a
general trapezoidal shape. The more recent Windows version MIDUSS 98 added an
option to allow the channel cross-section to be defined by straight segments joining up to
50 sets of coordinate pairs. (For details see Smith 1998).
However, the algorithm was intended for relatively compact channel cross-sections in
which the celerity can be approximated as 5/3 average velocity (from the Manning eq.).
With this simplification the resulting outflow hydrograph does not exhibit any of the
instabilities, volume loss or other anomalies discussed by the authors.
Following publication of the authors’ paper, the writer modified the MIDUSS 98 program
to allow the roughness coefficient (Manning ‘n’) to be specified for each segment of the
cross-section. This allows modelling of the type of compound channel examined by the
authors.
Figure 2 – Typical result from the Channel command
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When the ‘Variable roughness’ option is checked the conveyance of the channel is
computed as the sum of the conveyance of each segment of the cross-section. The
vertical division lines between all of the channel segments are assumed to have zero
roughness. The method is therefore similar to the ‘Vertical division method (VD)’
described by the authors and presumably suffers from the same drawbacks.
Figure 2 shows the results of the Channel command for the Ackers channel (p.598) with a
slope of 0.03% (S=0.0003) and with a peak flow of 100 c.m/s. Using the bed slopes in
the authors’ Table 2 the computed normal depth is shown in column 2 of Table 1 below.
It would be interesting to know if the methods used by the authors to subdivide the crosssection resulted in significantly different depths.
Table 1 – MIDUSS 98 results using asymmetric inflow hydrograph
(Qpeak = 100 c.m/s; Qbase = 10 c.m.s; x = 1000 m; t = 30 min)
Channel
bedslope
Normal
depth
(%)
(m)
0.3
1.959
0.1
0.05
0.03
2.495
2.916
3.282
MIDUSS98 outflow
Tpeak
Qpeak
Volume
(hours)
(c.m/s
loss (%)
18.0
99.865
<0.001
(18.5)
(99.70)
(0.86)
20.0
98.487
<0.003
(19.5)
(98.37)
(0.95)
21.5
94.189
<0.006
(21.0)
(93.96)
(0.47)
23.0
85.772
<0.018
(22.5)
(85.54)
(1.37)
The other columns in Table 1 show the result of applying the ‘Route’ command. For
convenience the authors’ results are shown below each value in parentheses.
For this experiment the Route command was modified to allow the number of subreaches to be modified to force the computed value of x to be changed. Also, the ratio
c/V could be adjusted to test the sensitivity of the results to the fixed wave celerity. The
number of sub-reaches was set to 20 to match the authors’ test. It was found that using a
value of c/V = 0.97 produced results acceptably close to those of the authors. The loss of
volume is negligible, the small percentage loss values shown in Table 1 being due mainly
to truncation of the hydrographs at 60 hours.
The four outflow hydrographs are shown in Figure 3 below. These do not display the
plateau in the rising limb that the authors obtained (Fig. 5). The authors suggest that this
should occur when the inflow reaches the bankfull value so that any increase in inflow is
absorbed in floodplain storage that does not contribute to outflow until the shallow wave
has traveled down the entire length of the floodplain. Simulation of this phenomenon
appears to require that celerity c is varied with flow Q. However, the authors’ computed
c = f(Q) curve (Fig. 3) appears to be computed by treating the compound cross-section as
a whole.
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Now for the data used for the authors’ Figure 3 and for a depth of 1.55m (i.e. a flood
plain depth of 50 mm) the celerity in the main channel is approximately 3.3 m/s whereas
the celerity on the floodplain is close to 0.25 m/s. Averaging the properties of the
compound channel the celerity is about 1.8 m/s. Also for channel depth y = 1.55 m the
flow in the channel is 60 c.m/s compared to an overbank flow of 0.5 c.m/s.
The question then arises as to whether the averaging process used to compute the
function c = f(Q) might affect the appearance of the plateau in the rising curve of the
outflow hydrograph. Do the authors know of any corroborative evidence - either
observed or simulated - to support this finding?
Figure 3 – Inflow and Outflow hydrographs referenced in Table 1.
In conclusion the authors are to be congratulated on extending the scope of the
Muskingum-Cunge method to treat the complex question of a compound channel crosssection with encouraging results. It is hoped that this discussion may encourage use of
both space and time weighting factors to allow spatial discretisation to be computed as a
function of the time step and the channel hydraulic characteristics. This may help to
reduce the instabilities in the recession limb.
References
Biesenthal, F.M. 1974 “A generalized approach to kinematic flood routing.” M.Eng
Thesis, McMaster University, Hamilton, Ont.
Smith,A.A. 1998, MIDUSS 98 User Manual – Chap.8, http://www.alanasmith.com