8/22/2019 fem5

1/7

A varying time step finite-element method forthe shallow water equationsC. Knock and S. C. RyrieFaculty of Computer Studies and Mathematics, University of the West of England, Bristol, UK

A vary ing time step predict or- corrector scheme, developed by Gresho et al. or t he Navi er-Sto kes equat ion sis applied t o the tw o-dimensional shall ow w ater equati ons. The equati ons, solv ed by inite elements usi ng theGalerki n method, are t ested against t he analyt ical soluti on to a one-dim ensional problem. They are also usedto model the Jlow w it hin t he Severn Est uary . The probl ems of selecti ng a suit abl e t im e-steppin g scheme, ofj i nding suitabl e boundary conditi ons, and of contr ol of accuracy are all di scussed.Keywords: finite elements, shallow water equations, time-stepping schemes

1. IntroductionMany finite-element codes have been developed and usedto model estuarine systems. 3 These codes use a varietyof spatial and time-stepping schemes that affect theaccuracy and cost. Hence the choice of scheme is veryimportant.

It is possible to use the finite-element method as thetime-stepping scheme, but Taylor and Davies3 found thatlinear finite elements in time damped the dominant waveform, and cubic elements in time, though extremelyaccurate, are too costly.

Finite-difference schemes are commonly used for thetime stepping as they are in general less costly thanschemes involving finite elements in time (see for instanceTaylor and Davies3). A wide variety of schemes are used,but the methods can be divided into two types, explicitand implicit methods. Explicit methods have theadvantage that they are cheaper per time step becausethey do not have to solve a nonlinear matrix as theimplicit methods do. However, the time step size, At, ofthe explicit schemes is limited by the condition

At < ~JS

where Ax is the element size, g is gravity, and h istotal water depth. Consequently explicit methodsgenerally require a greater number of time steps.Implicit schemes are generally not subject to thestringent stability criterion of the explicit method and

Address reprint requests to Dr. Knock at the present address: Centrefor Water Research, University of Western Australia, Nedlands, W.A.6009, Australia.Received 14 December 1993; revised 1 September 1993; accepted 25October 1993

hence can use larger time steps; Walters and Cheng4 useda fully implicit method for these reasons. However, animplicit method requires the solution of a nonlinearsystem for each time step, which is often carried out byNewton-Raphson iteration. Walters and Cheng found itnecessary to carry out two or three iterations per timestep to give a maximum change of less than 1% betweensuccessive iterations. This greatly increases the cost ofthe implicit schemes especially when using small timesteps, and so many authors use explicit schemes.5

More recently, systematic tests have been carried out.Cachet et al6 compared an explicit, three-pointpredictor-corrector scheme and an implicit, Eulerscheme. They recommend strongly the use of theimplicit scheme. For all problems where the domain isnot very large they found that the implicit methodperformed better than the explicit method due to the timestep size limitation of the explicit schemes.

Mathematical analysis of the one-dimensional linearshallow water equations using Fourier analysis has beencarried out by Foreman7** and Gray and Lynch.Foreman considered several methods on the basis ofability to predict the wave amplitude, phase velocity, andgroup velocity. He concluded that the better of thesecond-order schemes gave similar results to the onlysingle-step scheme, the Crank-Nicolson scheme, whichhe recommended as it is cheaper than the two-stepschemes.Gray and Lynch used Fourier analysis to analyze tenschemes. In the absence of friction they found that somemethods did not conserve mass and momentum, unlikethe Crank-Nicolson, leapfrog, split-step, and waveequation methods. The explicit method required thesolution of smaller matrices (which were symmetric) thanthe Crank-Nicolson method. However, when usingcomparable time steps the phase behavior of the leapfrogmethod was not as good as the Crank-Nicolson method.

224 Appl. Math. Modelling, 1994, Vol. 18, April 0 1994 Butterworth-Heinemann

8/22/2019 fem5

2/7

Time-stepping method for shallow water equations: C. Knock and S. C. RyrieThey also found that the Adams-Bashforth method hasunstable roots but may be useful for a short number oftime steps or if interspersed by a heavily damped timestep.

The time-stepping scheme used here was developed byGresho et al for the Navier-Stokes equations. It is animplicit predictor-corrector method with an automatic-ally varying time step and second-order accuracy in time.An implicit scheme was used rather than an explicitscheme because of the time step size limits of the explicitscheme. This means that the time step size depends onthe smallest element size, and in graded finite-elementmeshes this would be a severe restriction. The predictoris the explicit Adams-Bashforth method and thecorrector the second-order-accurate Crank-Nicolsonscheme recommended by several authors. The use of thepredictor means that only one Newton-Raphsoniteration is required for the implicit step, a considerablesavings in cost.The automatic variation of the time step size in theGresho algorithm gives an insight into the physics of theproblem. When the fluid flow is smooth a large time stepsize will give an accurate representation of the physicsbut a smaller time step size would be needed to followthe shedding of an eddy. In either case the code shouldautomatically choose the correct time step size.

2. Shallow water equationsThe finite amplitude shallow water equations, expressedusing Cartesian coordinates with z in the verticaldirection, consist of the continuity equation

dh ahu-_=at ax, n= 1,2and, neglecting atmospheric pressure and wind effects,the momentum equations

1 ahs$!+.,z+yh$+f.---m It ph ax,+ z, (bottom)ph = 0 n,m= 1,2

where h is the total water depth, rl is the water surfaceelevation, u, is the horizontal velocity, g is gravity, p isdensity of water, and f, is the Coriolis force.

The turbulent stress, T,,~, is given byrn_=v&+$)

where v, is the ttrbuleit viscosity, set as a constant inthis work. The bottom stress, r,, is given by

z, (bottom) gp V2ph c2

where C is the Chezy coefficient, assumed to depend onMannings coefficient, n, such that(Y/h

n

In this work we have chosen the value of n to matcha known tidal regime, as described in a later section.

2.1. Boundary conditionsWe will be solving the above equations in a domain

that represents a tidal estuary, and which thereforeincludes both an open (seaward) boundary and a closed(coastal) boundary. On a land boundary, the conditionused is that there should be no flow through theboundary, or that the flow should be known in advanceif, for instance, a river outfall is present. We make noattempt to model the movement of a coastal boundary,although in many circumstances this movement can besignificant,

The number and type of boundary conditionsappropriate on a seaward boundary are less obvious. Thenumber of conditions required for the depth-averagedshallow water equations are examined by Verboom etal 1 They conclude that, for subcritical flow, twoboundary conditions are needed when the flow is into thecomputation domain and one boundary condition isneeded when flow is outward.The condition used here is the Riemann boundarycondition, derived from the Riemann invariants of theone-dimensional shallow water equations. Verboom etal. demonstrate that problems of numerical reflection areless likely to arise from this type of boundary conditionthan from the commonly used method of specifying thesurface elevation. A disadvantage of the use of theRiemann condition is the difficulty of obtaining suitabledata by which to determine values of the invariant atthe boundary. Velocity components both normal to andtangential to the boundary are required.On inflow the conditions used are

-ug - 2& = al(t) = cs.c - 24&--u. t = b,(t) = -@,.J-

and on outflow-2j.E - 2& = q(t) = l/frlJ - 2&J

where a, and b, are predetermined functions of timecalculated from the field values of velocity and totalwater depth, us and h,.

On the coastal boundary if the viscous term is set tozero (as in some test cases) then

-u.n = 0--is set. If viscous effects are included then

-u.n = 0 -u.t = 0-- --is set.3. Numerical methodThe equations of motion are solved by the Galerkinfinite-element method using the finite-element code FEATdeveloped by Nuclear Electric. The depth and surfaceelevation were approximated by linear functions and theother variables by quadratic functions. A mixture oftriangular and quadrilateral elements was used.

Appl. Math. Modelling, 1994, Vol. 18, April 225

8/22/2019 fem5

3/7

Time-stepping method for shallow water equations: C. Knock and S. C. RyrieThese choices were made in light of the experience ofother authors using various types of finite-elementschemes. WaltersI finds, for the shallow waterequations, that mixed (quadratic/linear) interpolation,when compared with quadratic/quadratic interpolation,reduces oscillations in water depth but increases those inthe velocity variable. The latter may, however, beremoved by using suitable values of viscosity. Mixed

interpolation is moreover cheaper, as fewer equationsneed be solved at each time step. A more detaileddescription of the application of the Galerkin method isgiven by Knock.14

4. Test case: A depression advancing into still waterTo test the accuracy of the code in solving a nonlinearproblem and to choose an appropriate value of E forfurther calculations the following one-dimensional testcase was carried out. The problem, as shown in FigureI, is the modelling of a depression advancing into stillwater, from left to right. At t = 0, the region x > 0consists entirely of still water of uniform depthh = ho.The simplified form of the shallow water equationssolved was

The time-stepping scheme used is that developed byGresho et al. for the Navier-Stokes equations. It is animplicit predictor-corrector method with an automatic-ally varying time step. The predictor is the second-order-accurate Adams-Bashforth method, and the corrector isthe implicit second-order-accurate Crank-Nicolsonmethod.As applied to the equation

dhat+

d(hu) _ oaxau au al~+hx+g~=O

The influence of the depression issetting the following values to calculateinvariant:dYz- Qthe scheme isYp.n+l = y + $ [(2+$+-&l]

andY""=y"+$(Q"+QP,"")

where () is the value at the previous time step, ()p is thepredicted solution at the current time step, and ()+I isthe corrected solution at this time step.The matrix equation in the corrector was solved byone step of a Newton-Raphson iteration. Gresho et al.found that one step was generally sufficient to give goodaccuracy when compared with the results of a fullNewton-Raphson solution, and our computations borethis out.The step size is automatically calculated at each timestep using an error analysis of the predicted andcorrected steps, and is given byAt+ = At

where I#+( is the time truncation error of theprevious time step and E, the required error for the nexttime step, is a parameter set by the user at the start ofthe run.The value of E can have a significant effect on theresults. If it is too small then more time steps than arenecessary will be used, thereby increasing cost. If it is toolarge then the theory underlying the error estimate maybecome invalid, and the nonlinear system may becomedifficult to solve. Gresho et al. find that a value of 0.001(representing a 1% relative error at each time step) wasnearly the optimum value, and this was borne out byour own computations.

modelled bythe Riemann2 2;lc, tUf= -__ 3 on x = 0.0 m

where co = (gho)0.5.At the far end of the grid, the boundary condition ish, = ,/&, uJ = 0.0 on x = 8000.0m

with initial conditions u = 0 and h = ho. The valueschosen were ho = 10.0 m, H = 10.0 m, 1= 0.001 se1and g = 10 ms-. The maximum time of the run was800 s so that the depression did not have time to reachthe far boundary. Hence the simple boundary conditionh = ho could be used.The analytical solution to the problem is given byu(x, t) = 2(c(z) - co) h(x, t) = @!?#!9

where* = 1 + 2A - J(1 - nty + 41x/c,^.ZA.

Atc(z)co 3>Various meshes were used for testing, but eachconsisted of rectangular elements and was only oneelement wide as results were found to be constant acrossthe width of an element and the same results wereobtained for a mesh two elements wide as for a mesh one

EaUII=Om X- J000mFigure 1. Dimensions and mesh for the test case of a depressionadvancing into still water.

226 Appl. Math. Modelling, 1994, Vol. 18, April

8/22/2019 fem5

4/7

Time-stepping method for shal low water equat ions: C. Knoc k and S. C. Ryr ieelement wide. Hence for computational efficiency meshesonly one element wide were used.

ResultsA typical set of results on a 40-element mesh withE = 0.001 is shown in Figures 2 and 3. The absolute errors(differences between computed and analytic solutions)are shown in Figure 4. Two different types of errors werefound to occur. The largest error, E, in Figure 4, is definedas the amplitude of the maximum error of the result andoccurs where the depression enters the still water andthere is a singularity in the solution that cannot bemodelled exactly by the shallow water equations, givingrise to errors in the solution. The smaller error, E, inFigure 4, occurs in the error affected by the depressionand is defined as the magnitude of the oscillation.

To find the optimum value of E, the test case wasrepeated for various values of E and for varying element

Figure 2. Depression advancing into still water: Comparison ofresults and analytical solution for velocity with F = 0.001,Ax = 200 m, at (a) t = 200 s, (b) r = 400 s, and (c) t = 600 s.

Figure 3. Depression advancing into still water: Comparison ofresults and analytical solution for total water depth with E = 0.001,Ax = 200 m, at (a) t = 200 s, (b) t = 400 s, and (c) t = 600 s.

-0.0200 ( I I I I / I I0 1000 2000 3000 4000 5000 6000 7000 8000position (m)

Figure 4. Types of error that occur in the results of the testproblem of a depression advancing into still water.

Figure 5. Depression test problem: Cost vs. accuracy for (a) smallerror and (b) large error for constant Ax with E = 0.1, 0.01, 0.001,0.0001, and 0.00001 ; (c) small error and (d) large error for constantE with Ax = 100, 200, 400, and 800 m.size. Each test case was run for 400 s and at the end ofeach run the values of the two errors were calculated.Then the accuracy (defined as i/error) was plottedagainst the CPU time, or cost, for each run for varyingcases. Figures .5(u) and 5(b) show the results forconstant Ax and varying E and Figures .5(c) and 5(d)show the results for constant E and varying Ax.

For constant element size and varying values of E, thefigures show that the accuracy of the results increases asthe accuracy of the time-stepping scheme increases forvalues of epsilon down to E = 0.001. For smaller valuesof epsilon the accuracy decreases or remains approx-imately the same. Hence there seems to be no reason touse a value of epsilon smaller than E = 0.001 as not onlyare the results less accurate but the cost is greater.

The reason why the accuracy decreases for extremelyaccurate time-stepping could be that when the scheme isless accurate the errors are dispersed by numerical noisewhereas for the more accurate schemes they are notdispersed and may even be propagated. The exceptionto this trend is for the small error for the Riemannboundary condition where the error continues todecrease slightly but with a far greater increase in costfor only a small increase in accuracy.

Appl. Math. Modelling, 1994, Vol. 18, April 227

8/22/2019 fem5

5/7

Time-stepping method for shallow water equations: C. Knock and S. C. RyrieThe optimal values of E found here reflect those foundby Gresho et al. for the Navier-Stokes equation and as a

result the value of E = 0.001 is recommended and wasused for all test runs and simulations.

For constant E and varying element size, Figure 5shows that the small error always decreases as theelement size decreases, suggesting that it is generated bymesh error. So the criterion for mesh refinement woulddepend on the accuracy of solution required. The casefor this is less clear when the large error is plotted, Figure5, presumably due to an interaction between the mesherrors and time errors once the mesh errors have beenreduced to the same order as the time errors, whenAx = 200.0 m.

5. Test case: Severn EstuaryThe Severn Estuary in Britain has the second highesttidal amplitude in the world. The large tidal amplitudemeans that the estuary is well mixed and can be modelledby the depth-averaged shallow water equations.

The geography of the estuary is shown in Figure 6 andthe mesh and mesh interpolated topography in Figures7 and 8. There has been much discussion in the literatureon the choice of the position of the numerical seaboundary for estuary studies. Heapsi has a survey ofthe discussion for the Severn Estuary. The resultsindicate that for models of the proposed barrage nearWestern-super-Mare, the boundary should be at least400 km from the barrage,r6 and preferably at the edgeof the continental shelf.17 For this model the areacovered was limited by the available bathymetry dataand so the model stops at Tintagel. As this is far closerto the estuary than recommended, the numericalinfluence of the boundary is likely to be felt by the flowwithin the computation domain. The eastern limit of themodel was set just north of Avonmouth, and it wasassumed that there was no flow through thisboundary.

At the seaward boundary, the tidal amplitude, rl, andthe velocity, V, are taken to vary sinusoidally, i.e.,

2nt 2lrtr = lcos> v=v,os)

Figure 6. The Severn Estuary and Bristol Channel

Figure 7. Mesh used to model the Severn Estuary and BristolChannel.

Figure 6. Mesh interpolated topography of Severn Estuary andBristol Channel.

where lo and V, are the tidal amplitude and velocitiesrespectively, and To is the period of the tide, Theseexpressions are used to calculate the values of theRiemann invariant used for the seaward boundarycondition.

For simplicity, it was assumed that the flow wasperpendicular to the sea boundary on inflow, i.e., that

u.t =o_In the Severn Estuary the largest component of thetide is the principal lunar tide, M ,. The next largest

components of the tide are S, (the principal solar tide)and N, (caused by the elliptical nature of the moonsorbit around the earth). Their amplitudes are 35% and19% of the principal lunar tide respectively. As theprincipal lunar tide is the dominant tide in the estuary,To was set to the period of this tide.In the results presented the viscosity was set to zero.This caused no problems with spatial oscillations of the

228 Appl. Math. Modelling, 1994, Vol. 18, April

8/22/2019 fem5

6/7

Time-stepping method for shallow w ater equati ons: C. Knock and S. C. Ryri etotal water depth, though there was probably enoughdamping from the bottom friction term to avoid suchoscillations.

The model was run with initial conditions correspond-ing to zero velocity and zero surface elevationthroughout. As Figure 9 shows, within three-quarters ofa tidal cycle from the start damping due to bottom stressmeans that the results are the same as for the fourth tidalcycle. By comparison, 1.5 tidal cycles were needed byPartridge and Brebbia r9 for an implicit method.However, they used a fixed time step and found that anexplicit method using a smaller time step size settleddown more quickly. In fact both methods used byPartridge and Brebbia needed approximately 30 timesteps to settle down, only one or two more than neededfor the model of the Severn Estuary.

ResultsWhen compared with observed values of the M , tide theoptimum results were obtained using the tuned boundaryvalues of V, = 0.35 ms- and y10= 2.73 m with aManning coefficient of n = 0.037, Table 1.Looking at the errors in the solution while alteringMannings coefficient and the boundary values ofvelocity and surface elevation, it was found that theresults were most sensitive to Mannings coefficient. A1% change in Mannings coefficient gave a greaterchange in the solution than a 14% change in theboundary velocity or a 3% change in the surfaceelevation.

The value of the Manning coefficient used is acompromise that gives a tide rising too early and toohigh in the upper part of the estuary (e.g., at Avonmouthand Newport), but too late and not sufficiently high inthe lower part (e.g., at Tenby and Swansea). The valueof n = 0.037 is higher than that used in one-dimensionalmodels of the Severn Estuary: For instance, Ryrie andBickley used n = 0.027, and Binnie and Partners2iused n = 0.025.

However, our value is lower than that of thetwo-dimensional model of Peraire et al2 who used

Q,, \ ,,y- llfracombe 4

g-2 \ TG \ - Newport 1I ,,\\ -3 1 cx ~-- Port Talbot 11 -9A 5 llfracombe 1-4 1 5000 I5000 25000 35000 45000time from start of tidal cycle is)

Figure 9. Surface elevation at Newport, Port Talbot, andllfracombe during the first and fourth tidal cycles.

Table 1. Comparison of observed and computed values for theSevern estuary.Site Observed Computed

rl g ? g error it error gllfracombe 3.08 0.0 2.92 0.0 -0.16 0.0Tenby 2.62 8.0 2.65 8.6 0.03 0.6Swansea 3.15 11.0 3.09 13.9 -0.06 2.9Port Talbot 3.15 11 .o 3.12 14.7 -0.03 3.7Flat Holm 4.00 30.0 4.00 30.6 0.0 0.6Newport 4.13 36.0 4.17 35.6 0.04 -0.4Avonmouth 4.22 40.0 4.30 38.8 0.08 -1.2Observed values are for the M2 tide. The computed results are forlo = 2.73 m, V, = 0.35 ms-, n = 0.037. The water surfaceelevation, 9, is in meters and the phase, g, in degrees.

] -g I iLY I/

Figure IO. Results of the Severn Estuary model after (a) 1.25, (b)1.5, (c) 1.75, (d) 2.0 tidal cycles.

n = 0.0402. This difference may be due to the omissionfrom the model of Peraire et al. of the Coriolis effect.This effect has a far greater effect on the results thantuning the boundary conditions or Mannings coefficientand its absence was found to cause the tide to rise fartoo early on the northern shore.

The velocity vectors are shown in Figure 10 for 1.25,1.50, 1.75, and 2.0 tidal cycles after the start of the run.The results demonstrate the problem of setting thetangential velocity on the boundary to zero. In FiguresIO(c) and IO(d) as soon as the flow enters the mesh itchanges direction away from that set by the boundary.

This is also shown by the time step size for the run, asseen in Figure Il. When the flow changes from outflowto inflow the time step size drops and is controlled byerrors on the boundary. This suggests that the outwardflow does not match the flow now being forced into themesh, resulting in large differences between the predictedand corrected time step solution and hence a smaller timestep size. This demonstrates the need for accurate

Appl. Math. Modelling, 1994, Vol. 18, April 229

8/22/2019 fem5

7/7

Time-stepping method for shallow water equations: C. Knock and S. C. Ryrie

0 20000 60000 100000 140000 180000 220000time (s)

Figure 11. Time step size for the model of the Severn Estuary.

boundary data, a problem that does not always show upwith fixed time step models.

6. ConclusionThe above results for the test case and the Severn Estuarydemonstrate the successful use of the varying time stepGresho algorithm in the Galerkin finite-elementformulation of the shallow water equations. The resultsdemonstrate the need for a careful choice of the value ofthe time step control parameter and of accuracy controlboth in the grid and on the boundaries. They show theimportance of accurate field measurements for boundaryconditions: In some circumstances, the boundary datahave a controlling influence on the current time stepsize.

We have also confirmed some of the findings of thenumerical experiments of Gresho et al, relating to thesetting of a suitable value for the time step controlparameter E, and the need for the use of only one stepof Newton-Raphson iteration in the corrector step.

An important limitation in the practical application ofthis work is the lack of a model of the motion of theshoreline. The Severn Estuary, along with many others,has large areas of sand and mudflats, which are exposedat low tide. A suitable method for dealing with a movingshoreline, whether by moving elements or by specialshoreline elements, remains to be developed.

AcknowledgmentWe are grateful to A. G. Hutton of Berkeley NuclearLaboratories for much useful advice and assistance. Thispaper is published by permission of Nuclear Electric plc,who funded the work undertaken.

References1

2

3

4

5

6

I

8

9

10

11

1213

1415

16

171819

20

21

Werner, F. E. and Lynch, D. R. Field verification of wave equationtidal dynamics in the English Channel and Southern North Sea.Adv . W ater Resources 1987, 10, 115-130Peraire, J., Zienkiewicz, 0. C., and Morgan, K. Shallow waterproblems: A general explicit formulation. Znt . J. Num er. M efhodsEng., 1986, 22, 541-514Taylor, C. and Davies, J. M. Tidal propagation and dispersion inestuaries. Finit e Elements in Fluids, Vol 1, eds. R. H. Gallagher etal. John Wiley and Sons, New York, 1975, pp. 95-l 18Walters, R. A. and Cheng, R. T. A two-dimensional hydrodynamicmodel of a tidal estuary. Proceedings of t he 2nd. Int ernati onalConference on Fini te Element s in Wat er Resources, London, July1978Holz, K-P. and Hennhch, H. Numerical experience fromthe computation of tidal waves by the finite element method. Fin i teElements in W at er Resources, eds. C. A. Brebbia, W. G. Gray andG. F. Pinder. Pentach Press, London, 1976Cachet, J. F., Dhatt, G., and Touzot, G. Comparison of explicitand implicit models of tidal problems. Third InternafionalConference o n Fini te Element s i n Flui ds, Vol. 2, ed. D. H. Norrie,University of Calgary, Calgary, 1980Foreman, M. G. G. An analysis of finite element methods for thelinearised shallow water equations. Finite Elemenl Analysis, ed. T.Kawai. University of Tokyo Press and North Holland Publishing,pp. 541-554Foreman, M. G. G. An analysis of two time-step discretisationsin the solution of the linearised shallow water equations. J. Comp.Phys., 1983, 51, 454483Gray, W. G., and Lynch, D. R. Time-stepping schemes for finiteelement tidal model computations. Adv ances i n Wat er Resources,1977, 1, 83-95Gresho, P. M., Lee, R. L., Chart, S. T., and Sani, R. L. Solutionof the time-dependent incompressible Navier-Stokes and Boussi-nesq equations using the Galerkin finite element method. IUTAMSymposium on Approxi mati on M ethods for N avi er-StokesProblems, West Germany, September, 1979Verboom, G. K., Stelling, G. S., and Officier, M. J. Boundaryconditions for the shallow water equations. Engineering Applica-ti ons of Computat ional Hydraul i cs, Vol. 1, Pitman Press, Boston,1982, pp. 23G-262FEAT. FEAT-finite element computational analysis, User Guide,Berkeley Technological Centre, Berkeley, Glous, 1992Walters, R. A. Numerically induced oscillations in finite elementapproximations to the shallow water equations. Int. .I. Numer.M ethods Fl uid s, 1983, 3, 591-604Knock, C. Finite Element Modelling of Estuarine Hydro-dynamics. Ph.D. Thesis, Bristol Polytechnic, 1990Heaps, N. S. Prediction of tidal elevation: model studies for theSevern barrage. Severn Bar rage, Proceedi ngs of the Symposium ofthe Instit ute of Civ il Engineers, October 1981, pp. 51-58. ThomasTelford Press, LondonPrandle, D. Tidal Power Schemes in the Bay of Fundy and BristolChannel. Proceedings of the 2nd Int ernati onal Symposium on W aveand Ti dal Energy, Cambridge, September 1981Garrett, C. J. R. Tides in gulfs. Deep Sea Research 1975,22,23-35Uncles, R. J. Hydrodynamics of the Bristol Channel. MarinePollution Bulletin 1983, 15(2), 47-52Partridge, P. W. and Brebbia, C. A. Quadratic finite elements inshallow water problems. J. Hyd. Div. A.S.C.E. 1976, 102,1299-1313Ryrie, S. C. and Bickley, D. T. Optimally controlled hydro-dynamics for tidal power in the Severn Estuary. App. Math.M odell ing 1985, 9, 2-10Binnie and Partners. One Dimensional Model Studies of theSevern Estuary. Report No. WPTP(80)P345

230 Appt. Math. Modelling, 1994, Vol. 18, April