ANALYSIS OF INTERPOLATION STRATEGIES IN FIXED-GRID METHOD OF CHARACTERISTICS SOLUTION IN OPEN CHANNELS

Taban Sowlati and Bryan W. Karney

Department of Civil Engineering

University of Toronto, Toronto, ON, Canada, M5S 1A4

E-mail: karney@ecf.utoronto.ca

Tel: (416) 978-7776 Fax: (416) 978-6813

Abstract: The method of characteristics (MOC) has been widely used in the numerical computation of transient flow in closed conduits and open channels. This method, particularly in open channels and complex systems, has one significant drawback: exactly satisfying the Courant condition is impossible in almost all practical systems. Therefore, interpolation methods must be used and these inevitably produce physical and numerical errors. Numerical techniques associated with the use of the fixed-grid MOC are studied here in order to minimize the interpolation errors in transient open channel flow. Various methods are compared via two case studies: (a) a trapezoidal channel with an upstream reservoir and a downstream control valve and (b) a rectangular channel with a specified inflow hydrograph. Error is assessed through the mass balance, the simulated behavior and the time of occurrence of extreme depths for different grid sizes. The truncated spline can be used to avoid the tendency of conventional splines to over- or under-predict interpolated points; conventional cubic splines tend to have the lowest volumetric error for different grid sizes; and time-line methods tend to have consistent minimum depths at different levels of discretization. Results based on case (b) show that depth profiles obtained by cubic splines, and indeed for time-line interpolations, are consistent for different time steps and thus both techniques have the desirable property of grid insensitivity, unlike the space-line interpolation technique. Depending on the application, both cubic spline and time-line interpolations tend to perform well even with a relatively large time step, whereas space-line interpolation requires a smaller time step to obtain a similar accuracy. 

Keywords: unsteady open channel flow, interpolation techniques, method of characteristics

1    INTRODUCTION

Numerical modeling is a valuable and economical tool for predicting for flow variations in channels. Clearly, developing reliable and efficient mathematical and numerical models of the transient behavior of open channel flow is essential. One of the most challenging numerical problems is to determine the optimal numerical method and discretization strategy that will yield accurate, reliable and efficient solutions.

The governing equations for open channel transients are a system of non-linear partial differential equations based on continuity and momentum equations (Wylie and Streeter 1993) that cannot be solved analytically except in simple cases. The method of characteristics (MOC) is considered by some analysts to be one of the most suitable techniques for calculating unsteady flows in pipelines and channels (Chaudhry 1987) due to its simplicity and accuracy. Even though the MOC solution is stable, numerical damping and dispersion are inherent in the solution procedure (Goldberg and Wylie 1983, Ghidaoui 1993). In fact, any kind of interpolation produces a significant distortion in the physical problem. In fact, propagating waves cannot be tracked exactly and some kind of interpolation is unavoidable in complex systems.

Explicit interpolation schemes can be broadly divided into linear techniques, nonlinear techniques and, in case of closed conduit, the wave speed adjustment technique. Examples of nonlinear techniques include the Holly-Preissmann scheme (Holly-Preissmann 1977) and various spline schemes (Sibetheros et al. 1991, Ruan and Mclaughlin 1999). Linear explicit interpolation techniques are based on constructing a first-order polynomial between nodes. Examples of linear explicit schemes are space-line interpolation and time-line interpolation. Explicit interpolation methods estimate the unknown values of the dependent variables at the foot of the characteristic line from the known values at the other nodes. Logically, then, the first step of any interpolation procedure is to locate the point being interpolated. From the numerical viewpoint, the decision of where to locate the interpolated point and the specific choice of interpolation tends to control or limit the accuracy and practicality of a numerical scheme.

2    INTERPOLATION TECHNIQUES

Several interpolation techniques, namely space-line, time-line, cubic spline and truncated spline are introduced and compared here. A program, called DYNAMO (DYNAmic wave MOdel) written in the C programming language, calculates transient velocity and depth at a specified number of reaches in a channel section. Space-line and time-line approaches are based on linear interpolation in space and time direction, respectively. Cubic spline and truncated cubic spline techniques are based on non-linear interpolation in space direction. The cubic spline method with natural boundary condition ensures continuous first and second derivatives over the data set and has been recently used by other researchers. A truncated spline is introduced in DYNAMO in order to avoid the overshooting and undershooting associated with conventional spline techniques; a sensitivity factor is used to control the degree of truncation.

3    CASE STUDIES

The results for case (a) are based on a trapezoidal channel that is 1500 m long, 10 m wide, has a manning roughness of 0.016 and a bottom slope of 0.0002 (see Wylie and Streeter 1993). The side slope is 0.5 horizontal to 1 vertical. There is a reservoir at the upstream end and a valve at the downstream end. The simulated values of depth at the downstream end of the channel, based on three interpolation techniques, are shown in Fig. (1). The interpolation techniques used are cubic spline, space-line and time-line. The initial outflow in the channel is zero and is linearly increased to 40 m³/s. In case (b), the trapezoidal channel is 5000 m long, 10 m wide and has a specified hydrograph as a upstream boundary condition and a downstream reservoir. Detailed description of the case is given by Wylie (1980) and is not repeated here for the sake of brevity.

4    RESULTS

Fig. (1) shows downstream depth variation in time for case (a). Clearly, the results with 20 reaches are in close agreement, with the only minor differences at the minimum depth point. Downstream discharge variations based on space-line interpolation with 5 and 20 reaches are compared with cubic spline interpolation with 5 reaches for the second case study as shown in Fig. (2). The downstream discharge obtained by cubic spline with 5 reaches performs very similarly to the space-line interpolation with 20 reaches and the CPU time is also nearly the same (10.5 s and 10.3 s for space-line and cubic spline, respectively). Therefore, for a coarse discretization cubic spline interpolation is recommended. Fig. (3) is based on the second case studies in which inflow into the system varies in time. Channel outflow is simulated using cubic spline and time-line interpolation techniques for cases with both 5 and 20 reaches. The results for both techniques are almost the same for different discretization levels. The minimum scale for y axis is chosen as 35 m³/s so different profiles could be distinguished. Both techniques perform similarly with 5 reaches and also for 20 reaches.

5    ERROR ANALYSIS

One of the performance criteria considered in this paper for evaluating error introduced by each numerical technique is mass balance. Mass balance is a way to indicate whether or to what extent, the system under transients preserves the continuity law. Continuity balance C_B can be defined as (Wylie and Streeter 1993):  ; where  : initial volume stored in the channel, : final volume stored in the channel, : volumetric inflow at time t, : volumetric outflow at time t. Volume deficit (%) = is the base for continuity balance error analysis. Mass balance analysis has been done for space-line, time-line and cubic interpolation techniques in case (a). The results are tabulated in Table 1 based on 5 and 20 reaches. Not surprisingly, it was found that smaller reaches result in smaller mass balance errors. In any case, space-line interpolation has the largest volume deficit while cubic spline interpolation results in the smallest error. Two other criteria for evaluating errors are peak depth (min or max) and the time of its occurrence. Minimum depths and the corresponding times for all four interpolation techniques are tabulated in Table 2 for case (a) with different discretizations. It is shown that the space-line approach is the most is sensitive to the number of reaches. Its minimum depth varies from 1.95 m at 390 s with 5 reaches to 1.80 m at 520 s in 20 reaches. On the other hand, the time-line method is the least sensitive to discretization scheme, followed by truncated cubic spline method.

6    CONCLUSIONS

In transient flow computations, one particular interpolation scheme may not perform the best for all applications. This paper briefly explores various methods, such as linear interpolation techniques in space or time, and non-linear cubic spline or truncated spline interpolations applied to the modeling of open-channel transient flow. Two case studies are presented. The simplest method to implement is linear interpolation, although to achieve high accuracy a large number of reaches must be employed. Both time-line and cubic spline interpolation approaches exhibit high accuracy with larger time steps. Based on mass balance analysis, cubic spline introduces smallest error in the system for all discretizations level investigated here. It is also demonstrated that cubic spline technique with a coarse discretization performs as well as space-line interpolation with finer discretization and the computation times for both methods are nearly the same. Truncated spline is useful when there is an over or undershooting in the system.

References

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Ghidaoui, M.S. (1993). Analysis of Discretization Strategies in Fixed-Grid Method of Characteristics Solution in Closed Conduits. Ph.D. Thesis. University of Toronto.

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Holly, F.M., and Preissmann, A. (1977). “Accurate calculation of transport in two directions. ” J. Hydr. Engrg., ASCE, 103(11), 1259-1277.

Ruan, F., and Mclaughlin, D. (1999). “An investigation of Eulerian-Lagrangian method for solving advection-dominated transport problems.” Water Resources Research, 35(8), 2359-2373.

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               Table 1    Mass balance error analysis for case (a)

                 Table 2    Comparison of peak depths for case (a)

Fig. 1    Comparison of downstream depth variation using cubic spline, space-line and time-line interpolations for case (a) (N=20, Q1=0, QF=40cms, dt=10s )

Fig. 2    Downstream discharge using space-line N=20 (CPU time:10.49 s), N=5 (CPU time: 6.37 s) and Cubic spline N=5 (CPU time: 10.33 s) for case (b)

Fig. 3    Downstream discharge based on cubic spline interpolation for different grid sizes