Directory UMM :Data Elmu:jurnal:A:Advances In Water Resources:Vol23.Issue6.2000:

Advances in Water Resources 23 (2000) 645±651

A multiple grid algorithm for one-dimensional transient open channel
¯ows
Scott A. Yost *, Prasada Rao
Department of Civil Engineering, The University of Kentucky, 161 Raymond Building, Lexington, KY 40506-0281, USA
Received 22 November 1998; accepted 14 November 1999

Abstract
Numerical modeling of open channel ¯ows with shocks using explicit ®nite di€erence schemes is constrained by the choice of time
step, which is limited by the CFL stability criteria. To overcome this limitation, in this work we introduce the application of a
multiple grid algorithm to the ®eld of computational hydraulics. By coupling this algorithm to a second-order accurate MacCormack scheme, we demonstrate that the solution can be accelerated to the desired transient state. The present formulation has been
tested to simulate shocks arising from sudden closure of a sluice gate and for ¯ows accompanied with a hydraulic jump. The close
agreement between the obtained results and the theoretical ®ndings indicates the reliability of the proposed algorithm. Ó 2000
Elsevier Science Ltd. All rights reserved.
Keywords: Finite di€erence; Explicit; Transient; Shocks; Open channels

1. Introduction
Modeling ¯ow in open channels has drawn the attention of many researchers in the ®eld of hydraulic
engineering. As the ¯ow is often accompanied by shock
waves, formulating a reliable numerical scheme has always been a challenging task for the modeling community. To this end, the family of ®nite di€erence schemes,

owing to their ease of formulation, has found wide application. Based on the accuracy of the solution, the
explicit ®nite di€erence schemes can be classi®ed as either ®rst-order accurate or higher-order (P2) accurate.
For ¯ows with shocks, it has been well demonstrated in
the literature that the solution obtained by the second
and higher-order explicit schemes, although handicapped by the presence of oscillations in the vicinity of
the shock front, is much superior to the one obtained by
®rst-order schemes. A comprehensive review of various
®nite di€erence schemes can be found in the works of
Hirsch [1] and Chaudhry [2].
A characteristic feature of all the explicit schemes is
the choice of time step. From the stability perspective,
the magnitude of time step is limited by the CFL sta-

*
Corresponding author. Tel.: +1-606-257-4816; fax: +1-606-2574404.

bility condition [2]. For simulations over large time
periods, explicit codes using a small time step require
extra computational resources and additional computational time. Our motivation in doing this work is in
formulating an algorithm, which when coupled to any

®nite di€erence scheme, accelerates the solution to the
desired time period. The ideal properties that this algorithm needs to possess include (i) it should not a€ect
the accuracy of the solution, (ii) it should accelerate the
solution to the desired transient period, and (iii) its
formulation should be independent of the ®nite di€erence discretization. It is in this direction that multiple
grid methods look promising.
Multiple grid methods are numerical tools for accelerating the solution to the desired transient state. As
they are independent of the discretization approach used
for solving the ¯ow equations, their application is
quickly gaining momentum in many engineering ®elds.
To visualize the philosophy behind a multiple grid formulation, let us start the discussion by considering the
standard approach for arriving at transient pro®les. In
the standard and well-established approach, starting
from the initial conditions (i.e., time ˆ 0), the solution is
marched with time until the desired time period is
reached. Time marching is not accompanied by spatial
marching. By spatial marching we mean solving the ¯ow
equations more than once at any grid node. It is here

0309-1708/00/$ - see front matter Ó 2000 Elsevier Science Ltd. All rights reserved.

PII: S 0 3 0 9 - 1 7 0 8 ( 9 9 ) 0 0 0 5 2 - 4

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S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

that the multiple grid approach varies. In a multiple grid
formulation, time marching is accompanied by spatial
marching. At any time level, the solution is marched
over progressively di€erent grid levels (also referred to
as coarse grids), until the coarsest grid is reached. As
discussed during the course of this document, by spatial
marching we ensure that the ¯ow disturbance propagates to more than one node with a reduced computational e€ort. A review of the literature indicates that the
application of multiple grid techniques has been mainly
con®ned to problems requiring steady-state pro®les. Our
work di€ers from all the previously reported multiple
grid studies in that we formulate a multiple grid formulation for simulating transient ¯ows. This focus is
consistent with most free surface ¯ow simulations of real
life ¯ows, which are by nature transient. Examples include bores resulting from operating control gates at
power plant, rapid release of water from reservoirs and

spiked reservoir hydrographs.
As multiple grid methods for hyperbolic equations
are in their infancy, we hope this work will provide more
insights into their robustness. To test the eciency of
the formulated multiple grid technique, we tested it for
the most critical ¯ow combinations, ¯ows containing
moving shock fronts. The ®nite di€erence MacCormack
scheme was used as the basic numerical scheme for
solving the ¯ow equations given in Section 2. Sections 3
and 4 are devoted to the numerical scheme and the
stability condition. In Section 5, the multiple grid
strategy, its relation to the physics of the problem and
di€erent implementation details are discussed. The results of the present methodology for the two representative ¯ow cases are presented in Section 6.

2. Governing equations
For a horizontal rectangular channel, the basic governing equations based on the continuity and momentum principles can be written in conservation form as [2]
oh oq
‡
ˆ 0;
ot ox

oq o
‡
ot ox



q2 gh2
‡
h
2

…1†


m 2 uj uj
:
R4=3

In the present investigation we selected the MacCormack scheme to numerically solve Eqs. (1) and (2).
As the application of MacCormack scheme to open

channel ¯ows is well documented [2], we brie¯y describe
it to maintain continuity in the text. Starting from the
initial time level (n), the solution at the new time level
…n ‡ 1† is obtained using a predictor and corrector
approach.
As per this, the ¯ow equations in discretized form at
any node i can be written as
Predictor step:
Dt
…qi‡1 ÿ qi †n ;
Dx

n
Dt q2i‡1
q2
‡ gh2i‡1 ÿ i ÿ gh2i
qpi ˆ qni ÿ
hi
Dx hi‡1
hpi ˆ hni ÿ

‡ ghDt…S0 ÿ Sf †ni :

…4†

…5†

Corrector step:
Dt
p
…qi ÿ qiÿ1 † ;
Dx
p

Dt q2i
q2iÿ1
p
2
2
c

ÿ ghiÿ1
‡ ghi ÿ
q i ˆ qi ÿ
hiÿ1
Dx hi

hci ˆ hpi ÿ

‡ ghDt…S0 ÿ Sf †pi :

…6†

…7†

The ¯ow variables at the new time are then computed as
hin‡1 ˆ 0:5…hni ‡ hci †;

…8†

qin‡1 ˆ 0:5…qni ‡ qci †:

…9†

A characteristic feature of all second- and higherorder accurate schemes lies in producing dispersive errors near the vicinity of a discontinuity/shocks [3]. These
errors, which are manifest in the form of oscillations,
need to be smoothed at the instant of their generation.
In the present work we have used the procedure originally suggested by Jameson et al. [4] to reduce numerical
oscillations. As per this approach, the two ¯ow variables
are smoothed as
n‡1
ˆ hin‡1 ‡ ni‡1=2 …hi‡1
ÿ hn‡1
† ÿ niÿ1=2 …hin‡1 ÿ hn‡1
hn‡1
i
i
iÿ1 †

ˆ gh…S0 ÿ Sf †:

…10†

…2†

In the above equations h is the ¯ow depth, q the speci®c
discharge, g the acceleration due to gravity, S0 the
bottom bed slope of the channel, and Sf is the frictional
slope, computed using ManningÕs equation
Sf ˆ

3. Numerical scheme

…3†

Here m is ManningÕs roughness coecient, u the depth
averaged velocity and R is the hydraulics radius.

n‡1
ˆ qin‡1 ‡ ni‡1=2 …qi‡1
ÿ qn‡1

† ÿ niÿ1=2 …qin‡1 ÿ qn‡1
qn‡1
i
i
iÿ1 †

…11†

where
 n‡1

h ÿ 2hn‡1 ‡ hn‡1 
i‡1
iÿ1
i






ni ˆ  n‡1 
 ‡ hn‡1 
h
‡ 2hn‡1
i
i‡1

iÿ1

…12†

and

ni‡1=2 ˆ l max…ni ; ni‡1 †:

…13†

S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

The solution obtained by Eqs. (10) and (11) is free from
numerical oscillations. The smoothing mechanism, as
can be seen by Eq. (12), is triggered only in oscillatory
regions. For regions where the ¯ow is uniform, the numerator in Eq. (12) goes to zero, leaving the solution
computed using Eqs. (8) and (9) unaltered. The parameter l in Eq. (13) is known as dissipation constant
which controls the degree of smoothing. Based on
trial and error, in this work we have selected its value to
be 0.6.

4. CFL stability condition
A characteristic feature of the explicit schemes is the
choice of time step, which is governed from the stability
criteria. The magnitude of time step, given by the wellknown CFL stability condition [5], can be written as
D t ˆ Cn

Dx
p ;
max…juj ‡ gh†

…14†

where Cn is the Courant number ( 6 1) and Dx is the grid
spacing. With known grid spacing and ¯ow variables,
the time step can be computed using Eq. (14). Physically, Eq. (14) ensures that the numerical domain of
disturbance should be at least equivalent to its corresponding physical domain.

5. Multiple grid algorithm
In the literature, the reader often ®nds the terms
multigrid and multiple grid used interchangeable.
However, there exists a fundamental di€erence between
these two methods, which is detailed here. Multigrid
methods have drawn the attention of many researchers
since Brandt [6] initially proposed them. The objective of
using a multigrid technique is in accelerating the convergence of the solution. In the standard multigrid applications (explained later) the residual error
…r ˆ B ÿ AX † from the ®ne grid levels is transferred to
the progressive coarser levels. Computing the residual
error is possible when a system of equations is solved
simultaneously, as in implicit formulations where the
linearized equations are written in matrix notation,
Ax ˆ B. The formulation and associated applications
relating to multigrid techniques can be found in the
works of McCormick [7] and Wesseling [8].
Since explicit ®nite di€erence formulations directly
compute the ¯ow variables at the new time level from
the known initial conditions, implementing a standard
multigrid formulation turns out to be cumbersome. Ni
[9] circumvented this problem by directly passing the
approximate solution of the equations (rather than
the residual error) to the coarser levels. This gave rise to
the term multiple grid. Multiple grid techniques, by and

647

large, are credited to the initial work of Ni [9]. As
mentioned previously, in a multiple grid formulation, at
a ®xed time period the solution is computed at di€erent
grid levels (i.e., it is marched with space). Starting from
the ®ne grids (spacing of Dx), the coarse grids can be
obtained by skipping the alternate ®ne grid nodes. Fig. 1
pictorially represents the grid levels along with the corresponding nodal spacings. The notations ÔfgÕ, ÔcgÕ and
ÔmgÕ represent ®ne grid, coarse grid and multigrid levels,
respectively, where the grid spacing increases by a factor
of 2 between each grid level. The ®nite di€erence formulation dictates that the wave front advances one grid
node at the end of one computation (one computation
implies applying Eqs. (8) and (9) at all the interior nodes
coupled with the associated boundary conditions). In
terms of wave propagation, advancing one node on the
ÔcgÕ level corresponds to 2 nodes on the ÔfgÕ level.
Advancing one node on the ÔmgÕ level corresponds to 4
nodes on the ÔfgÕ level. Since the computational algorithm is advanced sequentially (in space) from ÔfgÕ to ÔcgÕ
levels, at the end of one complete iteration the wave
front has advanced over 4 ®ne grid nodes. In the present
work, we used a one-to-one mapping when prolongating
the ¯ow variables from coarser to ®ner levels. This oneto-one prolongation mechanism is indicated by the arrows in Fig. 1. Formally the whole procedure can be
written as
(i) With the given initial conditions and computational domain, compute the time step using Eq. (14).
(ii) At this new time level, apply Eqs. (4)±(11) to
compute the values of ¯ow variables at all the interior grid nodes (ÔfgÕ level, spacing of Dx). Apply
appropriate boundary conditions at the end nodes.
(iii) Transfer the problem onto the next coarser level
(i.e., ÔcgÕ or ÔmgÕ levels). Compute the time step with
the new grid spacing (i.e., 2Dx or 4Dx)
(iv) Apply Eqs. (4)±(11) to compute the values of
¯ow variables at the coarse grid nodes with the modi®ed time step, and spacing.
(v) Repeat steps (iii) and (iv) until the coarsest grid is
reached.
(vi) Reassign all the computed ¯ow variables as initial conditions for the next time step; compute the
time step using Eq. (14) and go to step (ii).
The above-mentioned discussion is valid for both transient and steady-state ¯ows. Viewing it from a transient
perspective, Fig. 1 indicates that instead of advancing
over one grid node, the wave front has advanced over 4

Fig. 1. Schematic representation of grid levels and the prolongation
mechanism ( ± grid node).

648

S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

grid nodes at the end of one iteration with time (one
iteration is de®ned as one computation at every grid
level, i.e. steps (i)±(v)). Had only the ÔfgÕ level been used
in the solution, advancing the solution over 4 ®ne grid
nodes would have required 4 time steps. Extending this
discussion to the end of the second iteration, the wave
front advances 8 grid nodes, which conceptually would
only have been 2 ®ne grid nodes had the solution not
been transferred to any coarser grid levels. This indicates
that at the end of every iteration the wave advances
spatially by an additional 4 nodes.
Correlating the increased spatial advance on the ÔfgÕ
level to the period of simulation is a critical parameter in
this investigation: say, if the transient solution at time
period t is required, and the time step obtained using the
CFL condition (Eq. (14)) is Dt. With this information,
one can conclude that t/Dt steps are required to arrive at
this transient state. On the other hand, at the ÔcgÕ level,
since the grid spacing is doubled and hence the time step
(noting that Eq. (14) indicates that Dt is proportional to
Dx), t/2Dt steps are required. Extending this analogy to
the ÔmgÕ level, t/4Dt steps are required. It is this reduced
number of steps, at higher grid levels, which aids in
faster convergence of the solution to the desired transient state The results of our investigation show that this
approach is a reliable mechanism. Since the computations at the higher grid levels are based on the nature of
the ¯ow and the time step, which in turn is subject to
the stability condition, any error introduced by this
approach is minimal.
A critical aspect in the above formulation relates to
the maximum coarse level that can be reached while
ensuring no deterioration of the solution. To keep the
truncation errors to a minimum (as the truncation error
is proportional to grid spacing), it is ideal to start with a
small ®ne grid spacing so that the corresponding coarse
grid spacing does not pose computational diculties. As
the solution is accompanied by numerical oscillations,
the magnitude of damping constant should accommodate the required amount of smoothing at all the grid
levels. An insucient value of l (Eq. (13)) can result in
excessive oscillations, which can amplify with time.
The most important factor a€ecting the maximum
coarse grid level that can be used is the accuracy of the
solution (i.e., does the solution obtained using coarser
grids resemble the solution obtained by using only ®ne
grids?). Any computational saving could be lost if the
solution deteriorates (i.e., smeared shock) between a ®ne
grid and multiple grid computation. While reviewing the
robustness of ®nite di€erence schemes, Woodward and
Collela [10] noted that the length scale between nodes is
an important factor for capturing shock fronts. For
standard ®nite di€erence schemes, where the solution is
computed only on the ÔfgÕ level, one can theoretically
arrive at solutions with minimum smearing of the shock
front by choosing the grid spacing, Dx, to be very small.

The straightforward use of this small grid spacing involves a trade o€ between computational time and accuracy. Since the present multiple grid formulation
requires the solution to be marched with space, it is
plausible for the solution to deteriorate marginally for
formulations using very coarse grid levels. Quantifying
the grid spacing at the coarsest level so as to keep the
truncation errors to a minimum is not an easy task and
can require extensive numerical runs. Our experience
shows that for multiple grid formulations using up to
the current ÔcgÕ level (Fig. 1), not much ¯ow information
is lost. The results in the next section con®rm the reliability of this multiple grid method in resolving the
shock front consistent with the ®ne grid solution while
gaining computational eciency using coarse grids.

6. Application
6.1. Case 1: surge waves
As ¯ows with surges and shocks are considered to be
a critical test for code validation purposes, we have selected ¯ow scenarios that have distinct moving shock
fronts. The ®rst test case relates to waves arising from a
sudden closure of gate at the downstream end. The
de®nition sketch of the problem is illustrated in Fig. 2.
At time t ˆ 0‡ , the gate at the downstream end is instantaneously closed, resulting in a regressive elevation
wave propagating upstream. In the numerical simulation, the initial conditions correspond to a depth of 6 m,
a unit discharge of 3.125 m2 /s, a grid spacing (Dx) of 5 m
and a Courant number of 0.9. An important aspect in
any numerical implementation is in specifying the
proper boundary conditions. At the downstream end, a
zero discharge is speci®ed, and the ¯ow depth is computed using the positive characteristic curve of Eqs. (1)
and (2). This can be written as [2]
qniÿ1
:
p
n
iÿ1
‡
gh
n
iÿ1
h

hin‡1 ˆ hniÿ1 ÿ  qn

…15†

iÿ1

To compare the computed values with the analytical
solution, the channel is assumed to be smooth (i.e.,
roughness coecient ˆ 0). The analytical solution can
be obtained by solving the following equations [11,12]

Fig. 2. De®nition sketch for a negative surge.

S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

649

Fig. 3. Transient pro®le at t  354 s …Dx ˆ 5 m; Cn ˆ 0:9†: (a) normal view; (b) zoom view.

h0 u0
‡ h0 ;
c

…16†

s
…h0 ‡ h1 †gh1
‡ u0
cˆÿ
2h0

…17†

h1 ˆ ÿ

iteratively. For the given initial conditions h0 and u0 , the
analytical solution yields the depth of the surge,
h1 ˆ 8:66 m moving with a celerity, c ˆ ÿ7:06 m/s. Of
interest is the transient pro®le at t  354 s, at which time
the wave is at the midlength of the channel. Fig. 3 shows
the numerical solution corresponding to this time period
for di€erent grid levels. A ®ne grid spacing of 5 m and a
Courant number of 0.9 were selected. The time step
obtained on the ÔfgÕ level using Eq. (14) was 0.1389 s.
This indicates that to arrive at the required transient
period, 2548 steps are required. On the ÔcgÕ and ÔmgÕ
levels, the number of steps reduce to 1274 and 637, respectively. The close agreement between the three solutions indicates that no appreciable amount of
information is lost in the process of coarsening the grid
spacings and in prolongating the ¯ow variables back to
the ®ne grid. The oscillation evident near the surge is a
feature of the MacCormack scheme used and not related
to the multiple grid procedure. We have selected a dissipation constant l value as 0.6. Though selecting a
higher value would have helped the solution to be more
smooth [4], investigating the optimal value is beyond the
scope of this work. An alternative of using varying
values of l, depending on grid level, is a subject left for
further investigation. The relative error
!
Pno: of nodes
…hexact ÿ hcalc †2
iˆ1
no: of nodes
based on the ÔfgÕ, ÔcgÕ and ÔmgÕ computations is 0.28%,
0.34% and 0.43%, respectively. The numbers in the

parenthesis in Fig. 3 relate to the required CPU time. As
indicated the time required for arriving at this time period using the ÔfgÕ grid level is around 78 s. On the other
hand by using the ÔmgÕ level, the solution can be accelerated to the same transient period in 46 s. All the runs
were made on the HP Exemplar (http://spp.uky.edu).
This computational saving, coupled with the small relative error between the numerical and analytical pro®les, indicates that the present multiple grid
methodology can be used satisfactorily.
6.2. Case 2: Hydraulic jump
The second test problems relate to a hydraulic jump.
A hydraulic jump is formed whenever the ¯ow changes
from super-critical ¯ow …Fr > 1† to sub-critical ¯ow
…Fr < 1†, where Fr represents the Froude number. The
de®nition sketch of the problem is shown in Fig. 4. The
initial ¯ow conditions in the horizontal channel are a
depth of 0.05 m and a velocity of 2.1 m/s. These ¯ow
conditions give an upstream Froude number of 3.0. A
Courant number of 0.8, a roughness coecient of 0.004,
and a ®ne grid spacing of 0.3 m (i.e. the number of
nodes ˆ 300) were used. As supercritical ¯ow has an
upstream control, the ¯ow variables at the upstream
node were kept equal to the initial conditions. At the
down stream end a constant depth of 0.2 m was speci®ed

Fig. 4. De®nition sketch for hydraulic jump.

650

S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

and held constant for all time levels. The ¯ow discharge
at the downstream end was computed using the positive
characteristic equation, which can be written as [2]

pn ÿ n‡1


hi ÿ hniÿ1
qin‡1 ˆ qniÿ1 ‡ u ‡ gh
iÿ1

ÿ …ghDtSf †niÿ1 :

…18†

Fig. 5 is the transient depth pro®le at t ˆ 120 s, along
with the required CPU, for various grid levels. The time
step obtained using Eq. (14) was 0.1428 s, which gives
the number of steps at the ÔfgÕ level as 840. On the Ôcg'
and ÔmgÕ levels the number of steps reduce by a factor of
2, to 420 and 210, respectively. As no experimental results are available for these transient conditions, we used
the solution obtained using ®ne grid as the benchmark
solution. The close agreement between the shock pro®les
using di€erent grid levels reinforces the conclusion

drawn from Fig. 3. Finally based on this result, the e€ect
of the source term in the governing equations on the
multiple grid formulation is minimal.
To study the e€ect of the Courant number on the
resolution of shock, we re-ran the Fortran code by
varying its magnitude. For Courant number of 0.4,
Fig. 6(a) and (b) shows the transient pro®le. The trend
of the results shown in Fig. 6(b) is similar to the one
shown in Fig. 5(b). Hence, one can safely conclude that
the e€ect of varying the Courant number on shock resolution is minimal.
The e€ect of increased grid spacing on the shock front
is shown in Fig. 7. A ®ne grid spacing of 1 m was used to
generate this plot. The corresponding spacings on ÔcgÕ
and ÔmgÕ levels were 2 and 4 m, respectively. The plot
indicates that the shock pro®le obtained using coarse
grids is smeared over few grid nodes. This plot is con-

Fig. 5. Transient water surface pro®le for hydraulic jump …t  120 s; Dx ˆ 0:3 m; Cn ˆ 0:8†: (a) normal view; (b) zoom view.

Fig. 6. E€ect of Courant number on the shock front …t  120 s; Dx ˆ 0:3 m; Cn ˆ 0:4†: (a) normal view; (b) zoom view.

S.A. Yost, P. Rao / Advances in Water Resources 23 (2000) 645±651

651

Fig. 7. E€ect of grid spacing on the shock front …t  120 s; Dx ˆ 1 m; Cn ˆ 0:8†: (a) normal view; (b) zoom view.

sistent with the observations made by Woodward and
Collela [10], who note that for numerically arriving at
high resolute shock pro®les, one needs to use a small
grid spacing in numerical investigations. Note that this
aspect of solution is absent in Figs. 5 and 6 wherein the
®ne grid spacing is 0.3 m. At this stage, we are not aware
of any work that quanti®es the optimal grid spacing.
Since the optimal grid spacing also depends on the initial
and boundary conditions, a trial and error approach to
arriving at its optimal value appears to be the only
feasible approach. Once the spacing is ®nalized, one can
march the solution with space, arriving at the pro®les
indicated by Figs. 5 and 6, which we believe is acceptable to the modeling community. By choosing a grid
spacing lower than Dx on the ÔfgÕ level, our experience
shows that the solution with coarser levels is more
accurate, as expected.

7. Conclusions
In this work, we introduced a multiple grid algorithm
for one-dimensional open channel ¯ow equations. The
algorithm that we discussed during the course of this
document can be coupled to any explicit ®nite di€erence
scheme for accelerating the solution to the desired time
level. By iterating the solution over a series of grid levels,
a multiple grid algorithm aids in faster propagation of
¯ow information. The advantages of using this algorithm have been numerically demonstrated by using it in
conjunction with the widely used second-order accurate

MacCormack scheme. Its reliability has been demonstrated by simulating transient waves, which commonly
occur in many real life open channel ¯ows.
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