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On the numerical modelling of tidalmotion in Singapore StraitHin-Fatt Cheong a , M. H. Abdul Khader a & Chor-Juen Yang aa Department of Civil Engineering , National University ofSingapore , SingaporePublished online: 18 Jan 2010.

To cite this article: Hin-Fatt Cheong , M. H. Abdul Khader & Chor-Juen Yang (1991) On thenumerical modelling of tidal motion in Singapore Strait, Journal of Hydraulic Research, 29:2,229-242, DOI: 10.1080/00221689109499006

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On the numerical modelling of tidal motion in Singapore Strait Modélisation numérique des courants de marée dans le Détroit de Singapoure

HIN-FATT CHEONG Associate Professor, Department of Civil Engineering, National University of Singapore, Singapore

CHOR-JUEN YANG Research Scholar, Department of Civil Engineering, National University of Singapore, Singapore

SUMMARY The Grubert two-dimensional hydrodynamic model was modified to simulate tidal elevations and tide-induced velocities. A stability analysis of the Grubert computational scheme showed that the scheme is "stronger" in one direction. The above scheme was modified and the stability analysis shows that the modified scheme is unconditionally stable in both directions for one dimensional flow situations. Applica­tion of the modified scheme to the Singapore Strait yielded satisfactory results.

RESUME Le modèle bi-dimensionnel de Grubert a été modifié pour représenter les variations de niveaux et les courants engendrés par la marée. Une analyse de stabilité du schéma numérique de Grubert a montré qu'il était plus robuste selon une direction privilégiée. Le schéma ci-dessus a été modifié et l'analyse a ensuite montré qu'il était inconditionnellement stable dans les deux directions pour des conditions de courant monodimensionnelles. L'application de ce schéma au Détroit de Singapoure a donné de bons résultats.

1 Introduction

An explicit scheme which is commonly used in the hydrodynamic modelling is the leap-frog scheme which uses a centered difference in both distance and time. It has the advantage of being nearly of the second order of accuracy. This scheme is adequately described by Mahmood [4]. Another frequently used explicit scheme is the Lax-Wendroff scheme [4]. This scheme differs in numerical characteristics with the leap-frog scheme. It is generally more stable and can be used in computations of rapidly developing phenomena - shock waves and tidal bores. But the inherent dissipation may, however, have an undesirable effect in computations of the more sensitive system of layered flows.

Revision received March 19, 1989. Open for discussion till October 31, 1991.

JOURNAL OF HYDRAULIC RESEARCH, VOL. 29, 1991, NO. 2 229

M. H. ABDUL KHADER Associate Professor, Department of Civil Engineering

National University of Singapore, Singapore

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The explicit scheme suffers from a stability problem. Stability is tested by determining if a particular part of the solution, perhaps an initially unimportant part, is likely to grow without limit until it destroys the calculation. The stability of explicit schemes is often governed by the Courant-Lewy-Friederichs (CLF) condition. Perhaps, the most widely used implicit finite difference scheme is the one first proposed by Leen-dertse [3] in 1967. This scheme operates on three time levels, that is from n At to (n + l/2)Ar and then from (n + 1/2) At to (n + \)At where At = incremental time step. In the first step of computation, the momentum equation in the x direction and the continuity equation are solved implicitly using a double tridiagonal algorithm while the momentum equation in the ƒ direction is solved explicitly. In the second step, the momentum equation in the ƒ direction and the con­tinuity equation are solved implicitly using the double tridiagonal algorithm also while the momentum equation in the x direction is solved explicitly. The algorithm used is an alternating direction algorithm. It first makes all the sweeps in one direction with the other direction variables locked and then makes all the sweeps in this other direction with the other original direction variables locked. In the implicit scheme proposed by Abbott (described by Sobey [6]) which operates on three time levels also, the velocity component u in the momentum equation in the x direction is made to advance directly from n At to (n + l)A/ while advancing h, the water depth, in the continuity equation only from n At to (n + 1/2) At. In the second stage, v, the component of the velocity in the momentum equation in the y direction is made to advance directly from n At to (n + l)A? while advancing h in the continuity equation from (n + l/2)Ar to (n + l)Af. In this scheme, information is accumulated, during the cycle, on the h values only. Abbott and Marshall's [1] scheme is also a three-level finite difference scheme except that it uses a double tridiagonal algorithm, a single tridiagonal algorithm and an explicit operator to execute the calculation. However, this scheme uses a diffusive law approximation in the lateral direction. Grubert's [2] scheme operates on three time levels, using a double tridiagonal algorithm at both stages. The nonlinear terms udv/dx and vdujdy are time-centred by making two complete com­putations. Grubert concluded that there is hardly a difference in results whether the convective terms are time-centred or weighted on the (« + \)At time level. The advantage of implicit finite difference scheme is it is generally more stable. Leendertse [3] presented an analysis of the linearized finite difference scheme and showed that the system is unconditionally stable apart from a weak constraint from the Coriolis term. But this un­conditional stability was concluded too early because a weak instability still exists which in some cases becomes apparent only after many thousand of time steps or when a small friction param­eter is used. This arises because Leendertse did not consider the nonlinear terms when he performed the stability analysis. Weare [8] has shown that this instability is due to the imperfect time-centering of the nonlinear terms. The numerical experiments conducted by Weare confirmed a weak instability which is only controlled by the friction term in normal circum­stances.

2 Theoretical background

For long waves propagating in shallow water in which the water particle motion can be con­sidered as nearly horizontal, the governing equations, in terms of depth integrated velocities, can be written in the Eulerian form as Grubert [2]

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dh dh dh — + u — + v — + h dt dx dy

du dv dx dy

du du du dh — + u — + v — + g — = a dt dx dy dx

dv dv dv dh T~ + w — + v — + g — = b dt dx dy dy

= 0 (1)

(2)

a = CfV — gu (u2 + v^12

h Ci + gl* + K-viQa

b =

K =

h =

{u2 + v2)m

-Qu-gv hCl - + gly +

9 / N

dx

dy

Qh

Qh

(3)

Cwl^w—T —Auh)+—Auh) (4) h

d2 , , à2 . (uh) + — (uh)

dx

h

dy1

l ^ w l ^ w - ^ —Auh) + —1 (vh) (5) dx2 v ' dy2

(6)

(7)

Cc = the Chezy coefficient, Ct-= the Coriolis coefficient, Ec = the integration viscosity, g = the gravitational acceleration, Hb = the bed elevation measured from a fixed datum, h = the fluid depth, kw = the coefficient of wind shear stress, u and v = the depth integrated velocities in the x and y directions respectively, t/w and Vw = the wind velocities in the x and y directions respec­tively, Q and £>a = the densities of the water and air respectively.

3 Three-level implicit alternating direction finite difference scheme

In the formulation of the three-level implicit alternating direction finite difference scheme, Grubert [2] suggested a staggered grid system with the locations of the three variables h, u and v defined as shown in Fig. 1.

k+2

J-2 J - l

<- Ax ->

J+l J+2

® = location for the water depth ■ = location for the velocity

in the x direction | = location for the velocity

in the y direction Fig. 1. Staggered grid system.

Schéma numérique.

k+1

k

k - 1 Ay

i i ^ k-2

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Equation (1) and equation (2) are written in finite difference form for the first and second levels in respect of h for the three level implicit scheme. From equation (1), the finite difference equation can be written as:

. n + l/2 rn h" + U2 hn+U2 hn h" "j,k — "j,k "j+l,k — " j - l .k "j,k+l ~~ " j ,k- l — —+ u ——; -^+ v — —

At 4 Ax 4 Ay n + l n + l n n

W i + t k — " i —1 k *i k + 1 — ^ i k—I ~

+ h^±lA UA+hJ*±l tL_L = 0 (g) 4 Ax 4 Ay

The superscript denotes the time level of the variable concerned and the subscripts denote the grid location with x mentioned first. Similarly from equation (2), the finite difference equation can be written as:

, . n + i . . n . . n + i i . n + i t i P , . p wj,k — uj,k Mj + l,k — " j - l , k "j,k+l — " j ,k- l — —+ u ——' —'-+ v — — At 2 Ax 2 Ay r.n+1/2 fcn + l/2

+ V u - * j - . . f c = f l (9) 2 Ax

Equation (1) and equation (3) provide the second and third time levels in respect oï h of the three-level implicit scheme. From equation (1), the finite difference equation can be written as:

j,n + l/2 i,n + l/2 j.n+1 i,n + l " j + l,k ~ " j - l . k "j.k+1 ~ "j.k-1

4 Ax 4 Ay „n + l „0+1

+ / ? v j , k + , -v j , k _ , = o ( ] o ) 4 Ax 4 Ay

From equation (3), the finite difference equation can be written as:

„n + l „n p p n+l n+l vj.k ~ vj.k Vj+l ,k~ v j - l ,k vj,k+l ~ v j .k- l

At 2 Ax 2A^ . n + l . n + l ''j.k+l ~~ " j ,k- l

A n + I "j,k

+ h

— " j .k

At

;;n w j + l,k "

1/2

+ U

./n w j - l ,k

2 Ay = b (11)

Grubert [3] suggested that the variables with superscript p and those without subscript should be centred at (n + 1/2)At by iteration. Equation (8) to equation (11) are reduced to tridiagonal forms so that they can be solved using double tridiagonal algorithm. The tridiagonal form for the second level (x-sweep) equations can be expressed as follows:

4 uf+\\ + Bt hitm + q «jLY,k = Dj (12)

A'hfrf + B'ulV + Cjhptf = D] (13)

while the tridiagonal forms for the third level (y-sweep) equations can be expressed as follows:

«kVJY+1+/M"k+1 + ykVJÏi, = ^ (14)

«k/fjï+i + / W + ylhf& = <t>\ (15)

Aj, A], BJ, B*j, Cj, Cj, Dj, D], ak, ak, /?k, /?k, yk, y[, <fik and <p[ are known coefficients.

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4 Stability analysis

Following Stelling [7], Miller's theorems (refer to Appendix) are used to study the stability prop­erties of the Grubert's [3] scheme. The stability of the Grubert scheme is examined by consider­ing its application to a one dimensional flow problem. The one dimensional shallow water equations in x-direction in finite difference form with friction terms included are:

j.n+1 t,n . " "■' r..n + l ,,n + l . ,,n ,,n 1 . lf_^_ r . n + 1/2 j,n + l/21 r, / izr \

4 Ax 2 Ax

and

<k+1 - < k + ^ [</,k - « # y + f ^ [*&? - ^-Vi2]+? ^ < = o (I?)

To consider the propagation of a single Fourier component of wave number 2n/L, where L is the wave length, it is assumed that the error terms are of the form h = ffel2KK/L and u = Uea'"i/L where i = ]/— 1. The amplitudes, H and U, are functions of time and wave number. The eigenvalue equations and the reduced polynomial can be derived and shown to be

f(X) =x\\ + VX + iU') + X\GX- U'1 + iU'{\ + Wx)] + A 2 ( - 2 - W x - iU') + X(Gx-iU') + \ (18)

f'(X) = A4 + A3(GX + iU') + A2(- 2 - Vx + iU') + X[Gx-U'2-iU'(\ + ^ x ) ] + l + ^ - / £ / ' (19)

MX) = ^ [ ^ ( 2 + ^x) + U'2] + X2{ WXGX + iU'[Wx{2 + Vx) - Gx + U'2]} + X[- Vx(2 + Vx) - U'2] + Vx C7X - / f/'Gx (20)

where rx = Af/(2 Ax), 0X = 2 sin (£x Ax), Wx = g | u \ At/(hC2), Gx = ghrxQ\\2, U' = urx6x and kx

is the wave number in the x direction. According to Miller's theorems,/(A) is a conservative polynomial if and only if/i(A) = 0 and d//dA is a Von Neumann polynomial. The function fx{X) will only be zero if U' = 0 and Yx = 0. When these conditions are substituted into dfjdX, equation (21) results.

d//dA = A3(4) + A2(3C7X) + A(- 4) + Gx (21)

The function d//dA is described in Fig. 2. Three real roots result but they do not all lie within — 1 <X< 1. Hence the scheme with £/' = 0 is unstable. Miller's second theorem states that/(A) is a Von Neumann polynomial if/i(A) is a Von Neu­mann polynomial and |/*(0) | > \f (0) |. For the condition U' =f 0, then /i(A) =̂ 0 and the above inequality yields

\\+Vx-iU'\>\ (22)

a condition which is always satisfied. We now have to ascertain whether ƒ, (A) given by equation (20) is a Von Neumann polynomial. In order to determine whether the roots of equation (20) lie in the stability region where | A | < 1, a computer solution is resorted to by varying the values of Gx, U' and Wx. The results are shown in Fig. 3. It can be noted that the scheme is highly unstable even with a large friction term.

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Fig. 2. Function of d//dA for different G„ values. Variation de d/'/dA pour différentes valeurs de Gx

An attempt was made to resolve the numerical instability by modifying the above scheme. The following modifications to the difference scheme were made. The convective term udujdx in the x momentum equation was centered at the (n + l/2)Ar time level instead of the (n + l)Ar time level and the time level of the term hdujdx in the second half mass equation was raised from n to n + 1. Neglecting the friction term, the continuity and momentum equations in finite difference form, after the above modifications, now become

h At , ] ,.n + l 1 u A t ri.n+1/2 . n + 1/2"! r, ';-' + 2A L"j+i,k-"j-i.kJ+ r—L"j+i,k - "j-i.k J = u 'îjlk — "j,k ■

' ^ u At

"j,k + g At

(23)

; ^ [BJ& - «ft» + ^,,k - H ^ J + 1 ^ [Ajtfj* - ^,'i2] = 0 (24)

The eigenvalue equations and the reduced polynomial can be derived and shown to be

/(A) = A4(l + iU'jl) + A3(2GX- U'2\2 + iU') + A2(- 2) + A(- Ul2\2 - /£ / ' ) + 1 - /{/'/2 (25)

ƒ*(A) = A4(l + / f/'/2) + A3(- U'2/2 + i U') +A2(- 2) + A (2 Gx - U,2\2 -iU') + \ - i U'jl (26)

ƒ,(A) = A2(2GX - t U'GX) -2Gx + iU'GX (27)

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Fig. 3. Stability region. Domaine de stabilité.

When U' = 0, /,(A) ± 0. Hence, Miller's second theorem will be adopted. For | / (0 ) | > \f (0) |, the condition of 11 — /' U'jl \ > 11 + i U'jl | is always satisfied. fx (A) is a Von Neumann polynomial because A = ± 1. Hence the modifications have made the scheme unconditionally stable for flow in the x direction. Similarly, the one dimensional shallow water equations in the y direction in finite difference form without external forces are:

h At i.n + 1 U" -l r u n + 1 u n + l j . v n i)n l - l -"j,k — "j,k + . Lvj,k+1 — v j ,k-l + vj,k+l — vj,k-lJ +

vA? 2Ay

[ A f ^ i - ^ i i + A j V i - ^ J - 0

vj,k — vj,k + « 7 Lvj,k+1 — vj,k-lJ + j . L"j,k+1 — «j,k-lJ — u

(28)

(29)

where ry = A//(2 A.y), 0y = 2 sin (fcy Aj') and /cy is the wave number in the y direction.

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The eigenvalue equations and the reduced polynomial can be derived and shown as:

ƒ (A) = k2(\ - V'2I2 + Gy + n.S V') + X(- 2 - V'2\2 + Gy-i V') + ( l - / K ' / 2 ) (30)

f\X) = A2(l + / V'jl) + X{- 2 - V,2\2 + Gy + i V') + (\-V'2l2 + Gy-i\.5V') (31)

/1(A)=A[(K'2 + 2Gy) + (Gy-F'2 /2)2] + [(Gy- V'2\2)2 - {V2 + 2Gy) + i V;{V'2 - 2Gy)] (32)

where Gy = ghr2yQ2\2 and V' = vry6y and ky is the wave number in the ƒ direction.

From Miller's second theorem, |ƒ*(0) | > | ƒ (0) | or 11 — K'2/2 + Gy-/l.5V'\ > | 1 - i V\2 | which gives

2Gy+ V'2 + (Gy- V'2I2)2>0 (33)

a condition that is always satisfied. Besides this, J\(X) must be a Von Neumann polynomial. The root of the equation fi(X) = 0 can be shown to be

[(V2 + 2Gy) - (G y - r2/2)2] - iV'(V'2-2Gi)

(V'2 + 2Gy) + (Gy-V'2l2)2 ( }

For | A | < 1, it can be shown that

2 G y ( r 2 - 2 G y ) 2 > 0 (35) This condition is always satisfied since Gy is always non negative. It can now be seen that the Grubert's scheme is unconditionally stable in the y direction but not in the x direction. In order to illustrate the stabilities of the Grubert and the modified Grubert's scheme, a numer­ical experimental was conducted. Consider a one dimensional open channel oflength 9.5 km with a uniform mean water depth of 10 m and with the boundary conditions: u = 0 at x = 0 and Ç = 0.5 cos {(at) at x = 9.5 km in which £ is the elevation, co is the angular velocity (= 271/T) and T is the period with T = 25 At. The initial conditions are taken to be u = 0 and C = 0.5 m everywhere. The grid spacing is fixed at Ax = Ay = As = 200 m. Test computations were made using various values of At and Chezy coefficients corresponding to values of [i = At ^(gh)j(2As) = 1, 2, 4, and 8 and values of A = C2hl(2g As) = 8,16, 32, 64 and oo. Table 3 shows the experiment conditions. The results by the present numerical experiments for 500 time steps show that the scheme is stable in limits 8 < C2/i/(2g Ax) < œ and

2 Ax

The numerical experiments confirmed the above stability analysis and that Grubert's scheme is unconditionally stable in the.y direction but not in the x direction and the modifications done on the Grubert's scheme have made it unconditionally stable in the x direction.

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Table I. Numerical experiment conditions

G = 0.0314 and U' = 0.0253u for all cases /N X

T = 500s At = 20s C = CO c * = 0 X

T = 500s At = 20s C = 112 C * = 1.56 x 10"3u X

T = 500s At = 20s C = 79 C * = 3. 14 x 10"3u X

T = 500s At = 20s C = 56 C * = 6.26 x 10"3u X

T = 500s At = 20s C = 40 c * = 0.0123u X

T = 1000s At = 40s C = 0 0 c * = 0 X

T = 1000s At = 40s C = 112 C * = 3.12 x 10"3u X

T = 1000s At = 40s C = 79 C * = 6.28 x 10"3u X

T = 1000s At = 40s C = 56 C * = 0.0125U X

T = 1000s At = 40s C = 40 C

* = 0.0245u X

T = 2000s At = 80s C = co C * = 0 X

T = 2000s At = 80s C = 112 C * = 6.24 x 10"3u X

T = 2000s At = 80s C = 79 c * = 0.0126U X

T = 2000s At = 80s C = 56 C

* = 0.025u X

T = 2000s At = 80s C = 40 c

* = 0.049u X

T = 4000s At = 160s C = co c * = 0 X

T = 4000s At = 160s C = 112 C * = 0.0125U X

T = 4000s At = 160s C = 79 C

* = 0.0252u X

T = 4000s At = 160s C = 56 C * = 0.05u X

T = 4000s At = 160s C = 40 C

* = 0.098u X

1 2 4 8

5 Models' applications and discussion of results

The two dimensional hydrodynamic model presently developed is used to simulate tidal motion in Singapore Strait for the period from 5th August 1978 to 19th August 1978. The Singapore Strait model covers an area of 109 x 71 km2 (see Fig. 4). The orientation of the model is - 7.5 degrees relative to the geographic north. The lower left corner of the model boundary is approximately at latitude 1°2'30" N and longitude 103°20'0" E. It stretches from Pulau Pisang in the Malacca Strait in the west to the Horsburgh Lighthouse at the mouth of Singapore Strait in the east. In the south, the boundary cuts across the Riau Islands of Indonesia, while in the north, the Johore State of Malaysia. All relevant input data are obtained from the Port of Singapore Authority (PSA). These include the seabed topography or bathymetry below the lowest Astronomical Tide level, tidal elevations

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Bathymétrie depths are in feet below Chart Datum Mean sea level =1 67m

TIDAL STATIONS: CURRENT STATIONS:

Grid size in 1000 metres Fig. 4. Regional model discretization and bathymetry.

Discrétisation du modèle régional et sa bathymétrie.

with respect to the mean water level at the four open boundaries, grid locations of the tidal and current stations and their time history variations in tidal heights, current speeds and current directions from 5th August 1978 to 19th August 1978. The locations of tidal and current stations are given in Fig. 4. Simulation starts from calm conditions with the current speeds everywhere in the region set to zero. Simulation becomes stable after about two tidal cycles of simulation. Model calibration was done by adjusting the Chezy coefficient. Adjustment of Chezy coefficient led to negligible change in tidal elevations but significant changes in the current speeds and their phases. Final adjustment of Chezy values yield the results as given in Fig. 5 to Fig. 10. The results show that the tidal elevations are better simulated than the current speeds and current directions. For the comparison of the tidal elevations among the tide stations, the biggest phase lag is at T2 station (Fig. 5) while the smallest phase lag is at T4 station (Fig. 5). The phase lag at T2 station is about 0.9 hour while stations T3, T6, and T7 recorded about 0.6 hour of phase lag (Fig. 6). There are small phase lags at stations T4 and T5. Tidal amplitudes match well for all the tide stations except for the T5 station. The computed amplitude at this station is 20% off the measured value for the tidal cycle between 10th to 22nd hour. For the comparison of current speeds, the peaks of the speeds are not well matched for C4 station (Fig. 7) possibly because it lies near the South West open boundary where the linear extra­polation of velocity at open boundary has been assumed which gives rise to inaccuracy. This

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Fig. 5. Fig. 6. Comparison between computed results and Comparison between computed results and field data for tidal elevation. field data for tidal elevation. Comparaison entre résultats de calcul et Comparaison entre résultats de calcul et mesures nature pour les niveaux de marée. mesures nature pour les niveaux de marée.

Fig. 7. Fig. 8. Comparison between computed results and Comparison between computed results and field data for current speed. field data for current direction. Comparaison entre résultats de calcul et Comparaison entre résultats de calcul et mesures nature pour les vitesses des courants mesures nature pour les directions des courants. de marée.

inaccuracy is especially prominent at the tidal cycle between 10th to 22nd hour. The speeds at C5 station (Fig. 7) are quite well matched except those at the tidal cycle between 10th to 22nd hour. Among all the current stations, C6 station gives the best simulation of current speeds. Simulated current directions, on the whole, fit the field data. Only at certain periods, the deviations are found to be large. These occur at the tidal cycle between 10th to 22nd hour for the C4 and C5 stations. There are hardly any significant differences in results between the Grubert's scheme and the modified Grubert's scheme. The above model was applied to the same region for the period starting from 1 July 1985 for verifi­cation purpose. Due to lack of field data, verification of results can only be done by comparing

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simulated results with tide table results. Figs. 8 and 9 show the verification results. There have been 1.3 hours of phase lags at T10 and Ti l stations as shown in Fig. 9. The current speed at C7 station (Fig. 10) deviate quite significantly between 6th to 16th hour. There is about 1 hour phase lag in the current speed at C9 station (Fig. 10). The peak values are 25% to 30% out. The locations of tide as well as the current stations are given in Table 6.

D 2 i 6 ( ID i; u 16 is ;] ;? Tire I N

Fig. 9. Comparison between computed result and tide table result for tidal elevation. Comparaison entre résultats de calcul et données de la table de marée pour les niveaux de marée.

Fig. 10. Comparison between computed result and tide table result for current speed. Comparaison entre résultats de calcul et données de la table de marée pour les vitesses de courants.

6 Conclusions

Stability analysis of the Grubert's hydrodynamic scheme reveals that the scheme is weak for the flow in the x direction. This instability is mainly due to the inclusion of non-linear terms such as the convective term in the scheme. Although two complete iterations suggested in the Grubert's scheme will somehow stabilize the scheme, it is quite time consuming and the advantage of

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implicit scheme will be deprived. The modified Grubert's scheme has been proven to be un­conditionally stable both theoretically and experimentally for one dimensional flow situation. Whether it is unconditionally stable in two dimensional flow problem is yet to be found out because there are coupling terms such as udv/dx and vdu/dy which were not included in the stability analysis. Good agreement of the present hydrodynamic scheme results with field data shows that the hydrodynamics model presently developed works well.

Notations

Cc Chezy coefficient ( m ' / 2 / s ) C( Coriolis coefficient (s~') Ec integrat ion viscosity (m 2 / s ) g gravitational accelera t ion (m/s 2 ) Gx ghr\&\\2 Gy ghr)Q\\2 H bed elevation measured from a fixed datum (m) hb fluid depth (m) /x bed slope in the x direction /y bed slope in the y direction j notational index in the x direction k notational index in the y direction kx, ky the wave n u m b e r in the x and y d i rec t ions respectively (mT1) kw coefficient of wind shear stress L wave length (m) n notational index in the number of time steps rx A//(2Ax) (s/m) ry M 1(2 Ay) (s/m) T wave period (s) u depth integrated fluid velocity in the x direction (m/s) U' urx0, Uw wind velocity in the x direct ion (m/s ) v depth integrated fluid velocity in the>> di rect ion (m/ s ) V' vry0y Vw wind velocity in the y direction (m/s) x , y , z orthogonal directions in the Cartesian coordinate system respectively 6>x 2 s in (kx Ax) 0 y 2 sin (ky Ax) V x g\u\M){hCl) ( wave amplitude (m) H At^(gh)l(2As) A C2

chl(2gAs) A root of a characteristic equation Q density of the fluid (kg/m3) ga density of the air (kg/m3) co angular velocity (rad/s) Af time increment Ax distance increment in the x direction Aj> distance increment in the y direction

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References / Bibliographie

1. ABBOTT, M. B. and MARSHALL, G., A Numerical Model of a Wide Shallow Estuary, 13th Congress of the International Association for Hydraulic Research, Vol. 3, 1969, pp. 61-67.

2. GRUBERT, J. P., Numerical Computation of Two Dimensional Flows, Journal of the Waterways Harbours and Coastal Engineering Division, ASCE, Vol. 102, No. WW1, Feb., 1976.

3. LEENDERTSE, J. J., Aspects of a Computational Model for Long-Period Water-Wave Propagation, RM-5294-PR, The Rand Corporation, May, 1967.

4. MAHMOOD, K. and YEVJEVICH, V., Unsteady Flow in Open Channels Vol. I, Water Resources Publica­tions, P.O. Box 303, Fort Collins, Colorado 80522, U.S.A., 1975.

5. MILLER, J. J. H., On the Location of Zeros of Certain Classes of Polynomials with Applications to Num­erical Analysis, Journal Institute of Mathematics and Its Application, No. 8, 1971, pp. 397-406.

6. SOBEY, R. J., Finite Difference Scheme Compared for Wave Deformation Characteristics etc., Tech. Memor. No. 32, US Army Corps of Engineers, Coastal Engineering Research Centre, Washington D.C., 1970.

7. STELLING, G. S., Improved Stability of Dronkers' Tidal Schemes, Journal of the Hydraulic Division, ASCE, No. HY8, Aug., 1980.

8. WEARE, T. J., Instability in Tidal Flow Computational Schemes, Journal of the Hydraulic Division, ASCE, Vol. 102, No. HY5, Proc. Paper 12100, May 1976, pp. 569-580.

APPENDIX

Miller's (1971) theorems

Consider a polynomial with real or complex coefficients c:

f(z) = c0 + clZ + ... + cnzn (A.l)

Another polynomial ƒ *(z) is defined as: n

ƒ*(*) = I cn-jZJ (A.2) j=0

in which c denotes the complex conjugate of c. The "reduced polynomial" f\(z) is defined as:

/ ] U) = (A.3)

Note that fx{z) is always one degree lower than ƒ (z).

Definition 1 - The polynomial ƒ (z) is a conservative polynomial when the roots of the equation ƒ (z) = 0 lie on the unit circle.

Definition 2 - The polynomial ƒ (z) is a Von Neumann polynomial when the roots of the equation ƒ (z) = 0 are in the open unit disc or on the unit circle.

Theorem 1 - The polynomial ƒ (z) is a conservative polynomial if and only if/] = 0 and d//dz is a Von Neumann polynomial.

Theorem 2 - The polynomial ƒ (z) is a Von Neumann polynimial if either (a) | ƒ "(0) | > | ƒ (0) | and ƒ, is a Von Neumann polynomial; or (b) /j = 0 and d/ /dz is a Von Neumann polynomial.

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