Muskingum-Cunge MethodExplained Victor M. Ponce and Janaina Da Silva20 February 2020

 Abstract. The Muskingum-Cunge method is reviewed with the aim of encouraging its wider acceptance and use in current hydrologic engineering practice. Its theoretical background and computational accuracy are reviewed and clarified. The method is recognized to be the only numerical analog of the diffusion wave equation based on a straight-forward, explicit, point-by-point computation, all-the-while featuring grid independence. No other flood routing method can claim this particular set of properties at this time.

1.  MUSKINGUM VS MUSKINGUM-CUNGE METHODS

The Muskingum-Cunge method is a method of flood routing that improves on the classical Muskingum method (Chow, 1959) by using physical-numerical principles established by Cunge to calculate the routing parameters (Cunge, 1969). By way of comparison, it may be affirmed that while the classical Muskingum method is hydrologic in nature, the Muskingum-Cunge method has a distinct hydraulic basis. Table 1 compares both methods, describing their differences.

 Table 1.  Comparison between Muskingum and Muskingum-Cunge methods. Criterion Muskingum Muskingum-Cunge Date of origination 1938 1969 Original reference McCarthy, G. T. 1938. "The unit hydrograph and flood routing," unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24. Cunge. J. A. 1969. "On the subject of a flood propagation computation method (Muskingum Method)," Journal of Hydraulic Research, Vol. 7, No. 2, 205-230. Standard reference Chow, V. T. 1959. Open-channel Hydraulics, Mc-Graw Hill, New York, p. 607. Ponce, V. M. 2014. "The Muskingum-Cunge method," Section 10.6 of Fundamentals of Open-channel Hydraulics, online textbook. Nature Hydrologic method, where the routing parameters K and X are calculated by trial-and-error calibration, using a pair of inflow-outflow hydrographs measured in a gaged river reach. Hydraulic method, where the routing parameters K and X are based on the channel morphology, as represented by the prevailing channel slope and cross-sectional-shape characteristics. Ease of application depends on The availability of measured flood hydrographs suited for parameter calibration. The availability of geometric and hydraulic channel data, to the extent that this data properly represents the reach under consideration. Ease of computation Good with any computational tool, including a spreadsheet, programming, and available commercial and government software. Good with any computational tool, including a spreadsheet, programming, and available commercial and government software. Accuracy Good for the reach and event used for calibration; poor for any other reach or flood event. Very good for all events within the reach under consideration; only limited to the extent that the geometric and hydraulic channel data must be representative of the reach. Nonlinear flood routing Very limited; usually not available due to large hydrologic data requirements. Available at the cost of additional complexity. Overall assessment Dated method; limited in accuracy when used for extensive watershed/basin routing. Newer method, featuring state-of-the-art hydraulic numerical modeling knowledge; ideally suited for extensive watershed/basin routing.

Given the facts described in Table 1, we posit that the Muskingum-Cunge method is altogether theoretically better than the Muskingum method. Thus, we encourage its wider use in U.S. and world hydrologic and hydraulic engineering modeling practice.

2.  MUSKINGUM-CUNGE AND FLOOD WAVES

The key to the understanding of the theoretical basis of the Muskingum-Cunge method is the recognition that the diffusion wave applies through a wider range than competing waves such as the kinematic and dynamic waves (Ponce and Simons, 1977). Kinematic waves do not attenuate, while most flood waves attenuate at least a little bit; on the other hand, dynamic waves attenuate too much and, therefore, do not represent flood waves in typical cases. Diffusion waves, lying in the midrange of attenuation, are, by far, the most applicable wave from the standpoint of practice.

This fact was recognized early by McCarthy in 1938 (Fig. 1) and later by Cunge in 1969. However, unlike McCarthy, Cunge tied the Muskingum method to the properties on the governing diffusion wave. In this way, Cunge was able to relate the method's routing parameters to the geometric and hydraulic characteristics of the reach under consideration.

Fig. 1  The Muskingum river near Marietta, Ohio.

Given the demonstrated simplicity of the Muskingum-Cunge method, particularly when compared to alternative hydraulic routing models, the former remains a strong candidate among the gamut of available routing models. This is particularly the case in view of the fact that hydraulic routing methods are generally unsuited for hydrologic applications involving intensive watershed/basin routing, where, clearly, the use of a much simpler method is advised. Thus, the Muskingum-Cunge method emerges as the only diffusion-wave routing model which is simple and accurate enough to be suited for large-scale hydrologic modeling applications. In the remainder of this paper, we andeavor to make the case for justifying this assertion.

3.  MUSKINGUM-CUNGE AND COMPUTATIONAL ACCURACY

The strength of the Muskingum-Cunge method is clearly its theoretical basis as an analog of the diffusion wave equation. Cunge realized that the Muskingum method and the kinematic wave model shared the same theoretical basis. Furthermore, Cunge was able to calculate the error of the first-order numerical scheme (i.e., the Muskingum method) and to tie this error to the hydraulic diffusivity of the diffusion wave (Hayami, 1951). This accomplishment paved the way for the calculation of routing parameters in terms of geometric and hydraulic variables, thus circumventing the need for the expensive and impractical streamgage measurements.

The Muskingum-Cunge method is accurate because of its strong theoretical basis. Its numerical properties, including stability and convergence, have been extensively documented both in theory and in practice (Ponce and Vuppalapati, 2016). The method is strongly stable and with excellent convergence properties for values of Courant number in the neighborhood of 1. This property of strong stability and excellent convergence all but assures its grid independence, i.e., the property of a numerical scheme to reproduce the same result, regardless of grid resolution. Competing (numerical) methods, including the kinematic wave, can be shown to lack grid independence, casting a cloud on their theoretical correctness (Ponce, 1986).

In summary, the Muskingum-Cunge method is the only numerical analog of the diffusion wave equation based on a straight-forward, explicit, point-by-point computation, all-the-while featuring grid independence. No other flood routing method can claim this particular set of properties at this time.

4.  ROUTING EQUATIONS

The basic routing equation of the Muskingum-Cunge method is the following (Fig. 2):

Fig. 2  Definition sketch.

 n + 1                 n + 1            n             n            Q j + 1     =   C0Qj       +  C1Qj   +   C2Qj + 1 (1)

in which j is a spatial index, n is a temporal index and C0, C1 and C2 are calculated as follows:

 Δt - 2KX C0 = __________________            2K (1 - X ) + Δt (2)

 Δt + 2KX C1 = _____________________            2K (1 - X ) + Δt (3)

 2K (1 - X ) - Δt C2 = _____________________            2K (1 - X ) + Δt (4)

The parameters K and X are calculated as follows (Cunge, 1969; Ponce, 2014):

 Δx K  =  _____            c (5)

 1                   q    X  =  ___ ( 1  -  __________ )           2              So c Δx (6)

in which Δx = reach length (space interval); c = flood wave celerity; q = unit width discharge; and So = channel bed slope.

5.  LINEAR VS NONLINEAR ROUTING

In Nature, flood waves exhibit a nonlinear behavior, i.e., their celerity and attenuation properties (tend to) vary with the flow. The magnitude of this effect varies with the cross-sectional shape in predictable ways. Three asymptotic cross-sectional shapes are recognized (Ponce, 2014):

1. Hydraulically wide,

2. Triangular, and

3. Inherently stable.

The inherently stable channel is that for which the hydraulic radius is a constant in the overflow channel. The shape of the inherently stable channel has been calculated by Ponce and Porras (1995) (Fig. 3).

Fig. 3  The inherently stable cross-sectional shape.

It can be shown that the nonlinear effect is strongest for hydraulically wide channels, weak for triangular channels, and totally nonexistent for inherently stable channels. In practice, most cross-sectional shapes are likely to be close to being hydraulically wide. Thus, the nonlinear effect may be marked in certain, if not typical, cases.

Notwithstanding the nonlinear effect, there are two ways to calculate the routing parameters in Muskingum-Cunge routing (Ponce and Yevjevich, 1978):

1. The linear approach, and

2. The nonlinear approach.

In the linear approach, the routing parameters K and X are based on average (or representative) hydraulic variables (c and q) and kept constant throughout the computation. In the nonlinear approach, the routing parameters are allowed to vary with the flow, i.e., to vary for each computational cell as a function of local flow variables. The tradeoffs between linear and nonlinear Muskingum-Cunge routing are described in Table 2.

 Table 2.  Comparison between linear and nonlinear Muskingum-Cunge methods. Criterion Linear method Nonlinear method Main feature The routing parameters are kept constant throughout the computation The routing parameters are allowed to vary with the flow Mass conservation Zero (No loss of mass) Small but perceptible loss of mass Implemention Straight forward Somewhat difficult Field of application Engineering practice Research and specialized applications

6.  DISCUSSION

The Muskingum-Cunge method represents a considerable improvement in computational accuracy when compared with the related Muskingum method. The only caveat is that the parameter calculation (Eqs. 5 and 6) should be based on values of flood wave celerity c and unit-width discharge q that are representative of the reach under consideration. To acomplish this objective, it is recommended that GIS-supported geometric and hydraulic data be used to better estimate the relevant input data and variables on which to base the calculation of the routing parameters.

7.  SUMMARY

The Muskingum-Cunge method is reviewed to further clarify its theoretical basis and encourage its wider acceptance and use in current hydrologic engineering practice. Its theoretical background and computational accuracy are reviewed and clarified. The method is recognized to be the only numerical analog of the diffusion wave equation based on a straight-forward, explicit, point-by-point computation, all-the-while featuring grid independence. We note that no other flood routing method can claim this particular set of properties at this time.

REFERENCES

Chow, V. T. 1959. Open-channel Hydraulics. Mc-Graw Hill, New York.

Cunge, J. A. 1969. On the Subject of a Flood Propagation Computation Method (Muskingum Method), Journal of Hydraulic Research, Vol. 7, No. 2, 205-230.

Hayami, I. 1951. On the propagation of flood waves. Bulletin, Disaster Prevention Research Institute, No. 1, December.

McCarthy, G. T. 1938. The Unit Hydrograph and Flood Routing. Unpublished manuscript, presented at a Conference of the North Atlantic Division, U.S. Army Corps of Engineers, June 24.

Natural Environment Research Council. 1975. Flood Studies Report. Vol. 3: Flood Routing. London. England.

Ponce, V, M., and D. B. Simons. 1977. Shallow wave propagation in open-channel flow. Journal of the Hydraulics Division, ASCE, Vol. 103, No. HY12, December, 1461-1476.

Ponce, V. M., and V. Yevjevich. 1978. Muskingum-Cunge method with variable parameters. Journal of the Hydraulics Division, ASCE, Vol. 104, No. HY12, December, 1663-1667.

Ponce, V. M. 1986. Diffusion Wave Modeling of Catchment Dynamics. Journal of Hydraulic Engineering, ASCE, Vol. 112, No. 8, August, 716-727.

Ponce, V. M., and P. J. Porras. 1995. Effect of cross-sectional shape on free-surface instability. Journal of Hydraulic Engineering, ASCE, Vol. 121, No. 4, April, 376-380.

Ponce, V. M. 2014. Fundamentals of open-channel hydraulics. Online textbook.

Ponce, V. M., and B. Vuppalapati. 2016. Muskingum-Cunge amplitude and phase portraits with online computation. Online article.

APPENDIX:  Origin of the term Muskingum-Cunge

In a hydrologic context, the word "Muskingum" is taken from the Muskingum river, in eastern Ohio. The word paraphrases a similarly sounding Native American Delaware-language word, which some claim to translate as "Eye of the Elk." The term "Muskingum method" was first used by G. T. McCarthy in an unpublished manuscript dated 1938 and subsequently referenced by Chow (1959). McCarthy tested his newly developed hydrologic method of flood routing using data from the Muskingum river; hence the name.

 Jean A. Cunge

The word "Cunge" gives the proper credit to Jean A. Cunge, a Polish-French engineer who, in 1969, published the equations used in the Muskingum-Cunge method. The compound name Muskingum-Cunge was apparently first used in 1975 in the Flood Studies Report, published by the Natural Environment Research Council, London, England. In 1990, the Muskingum-Cunge method was incorporated into mainstream U.S. hydrologic engineering practice, in the U.S. Army Corps of Engineers' HEC-1 model, Version 4. In 1998, HEC-1 acquired a graphical user interphase capability (GUI) to become the HEC-HMS model.

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