A Multiquadric Solution for the Shallow Water Equations - CiteSeerX

A Multiquadric Solution for the Shallow Water Equations Yiu-Chung Hon1, Kwok Fai Cheung 2, Xian-Zhong Mao3 and

Edward J. Kansa4 ABSTRACT: A computational algorithm based on the multiquadric, which is a contin-

uously dierentiable radial basis function, is devised to solve the shallow-water equations. The numerical solutions are evaluated at scattered collocation points and the spatial partial derivatives are formed directly from partial derivatives of the radial basis function, not by any dierence scheme. The method does not require the generation of a grid as in the nite element method and allows easy editing or re nement of the numerical model. To increase the con dence on the multiquadric solution, a sensitivity and convergence analysis is performed using numerical models of a rectangular channel. Applications of the algorithm are made to compute the sea surface elevations and currents in Tolo Harbour, Hong Kong during a typhoon attack. The numerical solution is shown to be robust and free of spurious oscillations. The computed results are compared with the measured data and good agreement is indicated.

INTRODUCTION Numerical models based on the shallow-water equations are commonly used to predict sea surface elevations and currents due to tides and storm surge. In particular, the nite element method has been widely used in these models because of its exibility in modeling irregular coastline and varying bathymetry. Applications have been made to simulate tides and currents over extensive areas and promising results have been obtained (e.g., [1]{[3]). Asso. Prof., Department of Mathematics, City University of Hong Kong, Hong Kong. 2 Asst. Prof., Department of Ocean Engineering, University of Hawaii at Manoa, HI 96822, U.S.A. 3 Research Engineer, Zhejiang Provincial Institute of Estuarine and Coastal Engineering Research, Hangzhou, Zhejiang, P.R. China. 4 Research Physicist, Department of Earth and Environmental Sciences, Lawrence Livermore National Laboratory, CA94551-0808, U.S.A. 1

1

Although automated grid generation programs are available (e.g., [4]), the generation of a nite element grid with several thousand nodes and with elements of various sizes, shape and orientation is still a non-trivial task. In this paper, we examine the use of the multiquadric method (MQ) in solving the shallow-water equations. The method does not require the generation of a grid as in the nite element method and the computational domain is composed of scattered collocation points. It utilizes a radial basis interpolant to evaluate the partial derivatives of the numerical solutions which in turn are used in the time-integration procedure. The multiquadric method was rst introduced by Hardy [5, 6] to approximate topographic surfaces from scattered data points and applied the method to surveying and mapping problems. In Franke's [7] review paper, the method is rated as one of the best among 29 scattered data interpolation schemes based on the tests on accuracy, stability, eciency, memory requirement and ease of implementation. Stead [8] showed that partial derivatives obtained from the multiquadric interpolant are highly accurate making the method extremely useful in solving partial dierential equations. The multiquadric method has recently received attention for solving physcial problems in the form of dierential equations. Kansa [9, 10] derived a modi ed multiquadric scheme suitable for solving parabolic, hyperbolic and elliptic partial dierential equations. Golberg and Chen [11] combined the dual reciprocity method and the multiquadric method to approximate the particular solutions of partial dierential equations. Based on this multiquadric method, Hon et al. [12,13, 14] provided an ecient numerical scheme for solving various nonlinear initial- and boundary- value problems including Burgers's equation with Reynolds' number ranging from 0.1 to 10,000. However, the performance of the multiquadric method depends on the choice of a user-speci ed parameter r, which is often referred as the shape parameter. Golberg et al. [15] applied the technique of cross validation to obtain an optimal value of the shape parameter. In practice, the optimal value of the shape parameter can be determined by numerical experiments and the subject is still under intensive study by many researchers (e.g., [16], [17]). In this study, the value for the shape parameter suggested by Hardy [5] is used in the simulation. 2

To investigate the sensitivity of the multiquadric solution with respect to the shape parameter, time-step size and collocation point density, we linearize the shallow water equations by ignoring the wind stress, bottom friction, and coriolis force terms and de ne the problem on a rectangular channel with a constant depth so that the analytical solution can be used for comparison with the MQ approximation. To illustrate and examine the present method, the multiquadric model is applied to simulate the sea surface elevations and current in Tolo Harbour, Hong Kong during Typhoon Gorden in July 15-21, 1989. The computed sea surface elevations are compared with tide gauge data recorded by the Royal Observatory of Hong Kong and with the numerical results of a nite element model. Unlike the traditional nite element method, this MQ scheme does not require an upwinding technique to maintain stability as already shown in Kansa [10].

MATHEMATICAL MODEL In most circumstances, the sea surface elevations and current due to tides and storm surge can be described by the inviscid shallow-water equations. The ow is assumed to be vertically well mixed with a hydrostatic pressure gradient. In their primitive form, the depthintegrated governing equations include a continuity equation and a momentum equation in each of the x and y directions:

@ + @ (HU ) + @ (HV ) = 0 @t @x @y

p

@U + U @U + V @U + g @ + gU U 2 + V 2 = fV + x @t @x @y @x HC 2 Hw

p

(1)

(2)

@V + U @V + V @V + g @ + gV U 2 + V 2 = ?fU + y (3) @t @x @y @y HC 2 Hw where t = time; U , V = advective velocities in the x and y directions;  = sea surface elevation; h = mean sea level; H = h +  , the total water depth; C = Chezy coecient; f = coriololis parameter; g = gravitational acceleration, w = water density; x; y = wind 3

stresses in the x and y directions. The wind stresses x and y can be computed using the following formula: q

x = Wx Wx2 + Wy2 q

y = Wy Wx2 + Wy2

(4) (5)

where  = surface friction coecient and Wx and Wy = wind speeds in the x and y directions. The governing equations (1) to (3) are subject to appropriate boundary and initial conditions. The boundary conditions on the ocean and land boundaries are:

 =  (t)

(6)

Q~
~n = 0

(7)

where Q~ represents the velocity vector (U; V ),   is the speci ed sea surface elevation on the ocean boundary, and ~n is the unit outward normal vector on the land boundary. The initial conditions are:

Q~ = ~0

(8)

=0

(9)

which correspond to a still water condition and the mean sea level in the computational domain. This completes the de nition of the initial boundary-value problem. However, most of the nite element models based on the governing equations (1) to (3) exhibit spurious oscillations in the numerical solution and require the use of nonphysical dissipation (Gary [18]). More recent nite element models utilize the wave-continuity equation formulation introduced by Lynch and Gary [19], and a more stable and robust solution is obtained. Although the nite element method has become a useful tool in modeling tides and currents, the generation of a nite element grid for an extensive area can be tedious and time consuming. An alternative 4

method which does not require the generation of a grid is explored here. The algorithm developed can be applied more generally to solve other physical problems involving partial dierential equations.

NUMERICAL MODEL In this paper, the multiquadric method is applied to solve the shallow-water equations (1) to (3) subject to the boundary and initial conditions (6) to (9). A radial basis interpolant is used to interpolate the numerical solutions at scattered collocation points and compute their partial derivatives as needed in the time-stepping procedure. Contrary to the nite element method, the multquadric method utilizes a global interpolation and does not require the generation of a gird. To illustrate the numerical procedure, let F denotes the numerical solution of  , U , V or other quantities which are known at N distinct collocation points. The multiquadric method interpolate the scattered data using the following radial basis function:

F (~x) ' with and

N X

j q(~x; ~j ) +

(10)

j = 0

(11)

j =1 N X j =1

q

q(~x; ~j ) = (x ? j )2 + (y ? j )2 + r2

(12)

where j and = unknown coecients at time t; ~x = (x; y) = position vector of the eld point; and ~j = (j ; j ) = position vector of the collocation points. Here, r > 0 is the shape parameter controlling the tting of a smooth surface to the data. When r is small, the basis function ts a spiky surface to the data points. As r increases, the spikes at the data points spread out to form a smooth surface. When r is too large and reaches a critical value rcritical, the resultant matrix becomes ill-conditioned and the numerical errors increase dramatically. Note that the value of rcritical at a single precision machine is smaller than that at a double precision machine which indicates that the multiquadric method can provide a 5

highly accurate approximation using a high precision machine. Applying equation (10) at the N collocation points at which the numerical values of F at each time t are known gives the following system of linear equations: For i = 1 to N ,

F (~xi; t) '

N X j =1

j (t)q(~xi; ~j ) + (t)

(13)

As an interpolation problem, Micchelli [20] has proved that the system of equations (13) and (11) has a unique solution for distinct collocation points. Myers [21] extended this proof to a vector-valued function interpolation function based on the technique of kriging. Although the solvability of applying equation (10) as a collocation method to partial dierential equations is not yet derived, Kansa [10] believes that Micchelli's proof still holds because the solution of PDEs can be regarded as a special kind of interpolation. It is noted here that Fasshauer [22] proved the solvability of the collocation method applied to elliptic PDEs if Hermite basis approach is used. In matrix form, (10) and (11) can be written as 2 64

32

3 2

3

F 75 = 64 G 1 75 64  75 1T 0 0

(14)

where  is the vector for the N unknowns j , F is the vector with N elements F (~i), G is the N  N matrix with elements qij = q(~i; ~j ) and 1 is the 1  N vector of all ones. The rst partial derivatives of F with respect to x and y are given respectively as follows:

in which

N @F (~x; t) = X @q(~x; ~j )  j (t) @x @x j =1

(15)

N @q(~x; ~j ) @F (~x; t) = X  j (t) @y @y j =1

(16)

@q(~x; ~j ) = q (x ? j ) @x ((x ? j )2 + (y ? j )2 + r2

(17)

6

@q(~x; ~j ) = q (y ? j ) (18) @y (x ? j )2 + (y ? j )2 + r2 The partial derivatives of any order can be computed from the radial basis functions and the results are continuous over the computational domain. With the partial derivatives derived from equations (15) and (16), we have 2 64

@F @t

3 2 75 = 64

2 64

@F @x

3 2 75 = 64

2 64

@F @y

3 2 75 = 64

0

0

G 1 1T 0

32 3 75 64 _ 75 32

_

3

Gx 0 75 64  75 0T 0 Gy 0 0T 0

(19) (20)

32 3 75 64  75

(21) 0 where the overdot (_) indicates the time derivative and 0 is the 1  N vector of all zeros. In short forms, equation (13) can be expressed as

F (~x; t) = L(~x)(t)

(22)

@F = L_ @t

(23)

@F = L  x @x

(24)

@F = L  y @y

(25)

L = [ G 1 ]; Lx = [ Gx 0 ]; Ly = [ Gy 0 ]

(26)

with

where

and 7

2 3  = 64  75

(27)

L_  + H LxU + H Ly V + U (LxH ) + V (Ly H ) = 0

(28)

Equations (1) to (3) can now be discretized by using equations (23-25) to obtain

_U

L +(L Lx U )(

L Ly 

U )+(

V )(

U )+ g

Lx 

 +g

p

U 2 + V 2 LU = f LV + x 1T (29) HC 2 Hw

p

2 2 L +(LU )(LxV )+(LV )(Ly V )+gLy  +g UHC+2V LV = ?f LU + Hy w 1T (30)

_V

Therefore, we have h

_  = ?L?1 H LxU + H Ly V + U (LxH ) + V (Ly H ) _U



= ?L?1

"

(L Lx U )(

L Ly 

U) + (

V )(

U) + g

"

Lx 

 +g

i

(31)

p

#

p

#

U 2 + V 2 LU ? f LV ? x 1T HC 2 Hw (32)

2 2 _ V = ?L?1 (LU )(LxV ) + (LV )(Ly V ) + gLy  + g UHC+2V LV + f LU ? Hy w 1T (33) where  , U , V and H denote the unknown vectors [(t) (t)] at time t for  , U , V and H respectively. The matrix L depends only on the locations of the collocation points and hence the matrices Lx; Ly and L?1 are evaluated only once for a given set of collocation points. With a time integration scheme applied to equations (31) to (33), the matrix solution can be applied to interpolate the numerical solutions and to calculate their partial derivatives at dierent time steps. At each time step n, the numerical solutions on the boundaries are updated by the corresponding boundary conditions. The surface elevation  n is speci ed by the boundary

8

condition (6) and the current velocities U n and V n are computed from the boundary condition (7) as: U~ n = U n sin2() ? V n sin()cos()

(34)

V~ n = V ncos2() ? U n sin()cos()

(35)

where U n and V n are computed from the multiquadric interpolant and  is the angle of the outward normal ~n at the land-water boundary. To illustrate how to update the corresponding unknown vector  , we let = 0, (~i); i = (M + 1) to N , are points at the ocean boundary, L = G = q(~i; ~j ) = (qij ) is the constant N  N matrix,  = (i ) are the tide levels at points i, and (i) is the unknown vector . Let L?1 = (Qij ) be the inverse of the matrix L obtained by using Gaussian Elimination method with partial pivoting. At each time step n, we have

 n = Ln

(36)

Using boundary condition (6), we have, for i=(M+1) to N,

in =  

(37)

Let ~ n = (~i)n be the updated values of (i)n . We then have

~ n = L?1(1n ; 2n; : : : ; Mn ;  ; : : :;  )T

(38)

In other words, for i=1 to N, we have

~in = =

M X j =1 N X j =1

Qij jn + Qij

= ni +

N X k=1 N X

N X j =M +1

Qij  

qjk nk +

j =M +1

N X j =M +1

Qij (  ? jn ) 9

Qij (  ? jn) (39)

Similar formula to equation (39) can also be devised to update U and V at each time step to satisfy the land-boundary conditions (34) and (35). We also note here that in order to maintain stability, the maximum time step is chosen to satisfy
t  pd gh

(40)

where d is the distance between any two adjacent collocation points and h is the average water depth between the two points. The formulation of the multiquadric method is relatively simple compared to that of the nite element method. The method signi cantly reduces the preparation time for the input data and allows easy editing and re nement of the model. One drawback of the method is the choice of an optimal r for fastest convergence. The subject is still under intensive investigation. Recent numerical experiment by Tarwater [23] indicate that the errors usually drop to a minimum by increasing r to a certain magnitude and thereafter the errors grow enormously. This is also reported by Golberg & Chen [11], Bogomonly [24], and Cheng [25]. Carlson and Foley [16] found that the optimal value of r depends greatly on the magnitude of the numerical solution and has little in uence by the number and the locations of the interpolation points. Milroy et. al. [17] used the method of cross validation to adaptively search for the optimal value of r and obtained satisfactory results. While theoretical studies on the choice of the optimal shape parameter are still on-going, we adopt the suggestion from Hardy [5] by choosing r to be 0:815dmin , where dmin is the minimum distance between any two collocation points. Even though there is still no theory, ad hoc recipes suce if r  rcritical for convergence. In the following section, numerical examples are given to indicate the errors and convergence of the multiquadric solution for dierent values of r, time step size
t and number of collocation points.

SENSITIVITY Although the MQ has successfully been applied to provide solutions for various linear and nonlinear partial dierential equations, the method is applied here for the rst time to the shallow water equations. It is therefore necessary to verify the multiquadric model 10

with known solutions and to investigate the sensitivity of the model to the shape parameter, time step size and collocation point density. In order to compare with an analytic solution, the linear shallow water equations instead of equations (1) to (3) are used in the sensitivity analysis: !

@ + H @U + @V = 0 @t @x @y

(41)

@U + g @ = 0 @t @x

(42)

@V + g @ = 0 (43) @t @y The multiquadric algorithm is applied to a rectangular channel with length L = 872km, width W = 50km and depth H = 20m. The excitation wave period is taken to be 43,200 seconds (12 hours) and the wavelenght is calculated to be 605 km. The boundary conditions are:  (t) = 0cos!t on x = 0; 0  y  W

(44)

U (t) = 0 on x = L; 0  y  W

(45)

V (t) = 0 on y = 0 and y = W; 0  x  W

(46)

and the initial conditions are:

Q~ = Q~ (x; y; 0) = (U (x; y; 0); V (x; y; 0))

(47)

 =  (x; y; 0)

(48)

with 0 = 1m and ! = 1:45444  10?4 =s. The analytic solution to this boundary-value problem is known and is given by 11

 (x; y; t) = 0cos p!gH (L ? x) coscos!t p! L r g sin p! (L ? x) gH

gH

U (x; y; t) = ?0 H cos p! L sin!t gH V (x; y; t)  0

(49) (50) (51)

Since the wind stress, bottom friction, and coriolis force terms are not present, the solution corresponds only to the interaction between the incident waves and the re ected waves from the wall at x = L. Equations (31) to (33) are then simpli ed to h

_  = ?H (L?1Lx)U + (L?1Ly )V

i

_ U = ?g(L?1Lx)

(52) (53)

_ V = ?g(L?1Ly ) (54) We note here that the matrix products L?1Lx and L?1Ly are only evaluated once in equations

(52) to (54). For the time integration scheme, we apply the Runge-Kutta method of fourth order (RK4) to equations (52) to (54). In the sensitivity analysis, we simulate the tide  and currents U and V using the multiquadric model with N  5 collocation points, in which N = 21, 41 and 61 is the number of points along the length of the channel. As an example, gure 1 shows the arrangement of the collocation points for N = 21. For a time step size of 100 seconds, gures 2 and 3 show respectively the average relative errors of the calculated surface elevation  and the ow velocity U as functions of the shape parameter r ranging from 0:5dmin to 8dmin . The average relative errors for  are de ned with respect to the analytic solution as: RT Rl k Exact ? MQ kAve = 0 R0Tj RlExact ? MQ j dxdt 0 0

j Exact j dxdt

(55)

where T is the wave period and l is the channel length. The average relative errors for U is de ned similarily. The integrals in equation (55) are evaluated by the Simpson rule. From 12

gures 2 and 3, we have the same observation as Tarwater [23] that the average relative errors of  and U drop to a minimum as the shape parameter r increases to a certain value and thereafter the errors grow enormously. Figures 4 and 5 show the average relative errors of the surface elevation  and ow velocity U for N = 41 and the shape parameters between 0:5dmin to 6:5dmin . The results are generated for ve dierent time step sizes
t = 20, 50, 100, 200, 300 seconds. It can be observed from gures 2 to 5 that the MQ provides a stable and highly accurate approximation to the solution. It appears that an optimal solution is attained when
t = 20sec if the total number of collocation points N is 61. Since the sensitivity problem is linear, we can even apply a Laplace Transform technique to solve equations (54) to (56) which will completely eleminate the time integration error. The combine of this Laplace Transform technique with multiquadric method has recently been reported by Moridis and Kansa [26]. From the results, we can also observe that the proposed value 0:815dmin for the shape parameter r gives a reasonably good approximation to the solution. The numerical computations shows that the multiquadric scheme gives a very stable and accurate solutions. The y-direction velocity V is nearly zero. With increasing value of the parameter r, the errors drop to a minimum at around 4:5dmin for the case N = 61 and
t = 20s with an average relative error 0.05%. For r > 8dmin , the matrices becomes ill-conditioned that the solution cannot be achieved. This can be explained from the basis function de ned in equation (12) of the multiquadric formulation. If the shape parameter r is too small, the basis function is nearly linear. If the shape parameter r is too large, the basis function tends to a constant. In both cases, the multiquadric approximation fails to give a reasonable approximation. The convergence of the solution with respect to the number of collocation points and time step size is relatively rapid. For comparison with dierent order of time integration schemes, we apply also the Modi ed Euler method (MEM) of second order to equations (52) to (54). The average relative errors of  and U as well as the maximum relative errors de ned as

? MQ j k Exact ? MQ kMax= max j Exact  0

(56)

are listed in Table 1. The Runge-Kutta method is shown to provide a much better conver13

gence in comparison to the modi ed Euler method.

NUMERICAL EXAMPLE To illustrate the application of this method, the multiquadric tide model is also applied to simulate the sea surface elevations and currents in Tolo Harbour, Hong Kong during Typhoon Gorden in July 15-21, 1989. Figure 6 shows a map of the Tolo Harbour. The embayment of the Tolo harbour consists of a shallow inner basin (inner Tolo harbour) and a narrow and deep channel which extends into Mirs Bay (outer Tolo harbour) in the South China Sea. The embayment occupies an area of 50 km2 and is 16 km long. The width of the embayment varies from 5 km in the inner basin to just over 1 km at the entrance of the harbour. The inner Tolo harbour is shallow and is less than 10 m deep, while the Tolo channel is greater than 20 m deep. The tides in Tolo Harbour are mixed semi-diurnal with a predominant period of 24.5 hours. The overall tide range is from 0:1 m to 2:7 m. The computed sea surface elevations are compared with the tide levels recorded at Tai Po Kau and Ko Lau Wan located respectively in the inner and outer Tolo Harbours. The tide records are maintained by the Royal Observatory of Hong Kong. The computations are performed on a SUN Sparc 20 workstation with double precision for a four day period during the typhoon attack. A time step size of 30 seconds was used throughout the simulation. The input tide level at the ocean boundary is obtained from a tide gauge outside Tolo Harbour. The following equation suggested by the Royal Observatory of Hong Kong [27] is used to compute the tide level at the ocean boundary:

j(t) = a(t + TCORj ) + HCORj

(57)

where a(t) is the actual tide gauge record at time t, TCORj is the time correction parameter, HCORj is the tide level correction parameter and j(t) is the estimated tide level at point (xj ; yj ) and time t. In the computation, the coriollis parameter f = 0:5581e?4 , Chezy coecient C = 60, density of sea water w = 1, and wind drag coecient  = 0:16e?6. The computational domain is shown in gure 7 in which Tolo Harbour and part of Mirs Bay are modeled. A total of 302 collocation points are used in the simulation. There is 14

no speci c requirement for the arrangement of the collocation points. To compare with the nite element method, we generate a total of 346 triangular elements with 250 nodes for the same computational domain as shown in gure 8. The computed and measured sea surface elevations at Ko Lau Wan and Tai Po Kau are shown in gures 9 and 10 respectively. The numerical models reproduce the measured data very well. A small phase lag is observed between the computed and measured elevations, and is primarily due to the computation of the ocean boundary condition using an oshore tide gauge. The computed surface elevations are quite steady and free of the spurious oscillations as observed in some nite element models (Hon [28]) using the same governing equations to the same computational region. This is because the radial basis function in the multiquadric method is continuously dierentiable and hence can provide a much smoother approximation of the partial derivatives than the nite element method which uses low order polynomials as local basis functions. It is also observed that a larger time step (45 secs) can be used in the multiquadric model and still maintain stability whereas the nite element solution already diverges under the same settings. Plots of the ood and ebb current elds at time 26 hours 20 minutes and 35 hours 40 minutes are shown in gures 11 and 12 respectively. Only the velocity vectors at the collocation points are shown. Since there is no systematic monitoring of the currents in the Tolo harbour, the predicted current elds cannot be veri ed. However, previous eld measurements of the current ow in the channel recorded an average of 10 cm/sec, which is in consistent with the numerical predictions.

CONCLUSIONS The multiquadric method has been used as a spatial approximation scheme for the shallow water equations. The method utilizes the radial basis function to interpolate the numerical solutions at scattered collocation points and does not require the generation of a grid as in the nite element method. This substantially reduces the data preparation time and allow easy editing and re nement of the numerical model. The sensitivity of the multiquadric solution to the shape parameter, the size of time step and the number of collocation points 15

is investigated. The use of the shape parameter as suggusted by Hardy [5] appears to produce accuracy results in the numerical solution of the shallow water equations. The convergence of the solution with respect to the number of collocation points and time step size is also investigated and is found to be relatively rapid. To illustrate the method, applications are made to simulate the sea surface elevations and currents in Tolo Harbor during Typhoon Gordon in 1989. The computed results are compared with actual measurements and good agreement is indicated. The spurious oscillation found in some nite element models has disappeared and the computed results are stable and accurate. Although the resulting coecient matrix from the multiquadric method is of full, the matrix solution is required only once and the computational eciency can be improved by the domain decomposition method (Kansa [29]). The simplicity of the numerical formulation and the easy of implementation will surely make the multiquadric method an attractive alternative in solving various types of partial dierential equations. It is shown in this study that the multiquadric method leads to very accurate predictions with an unprecedent ease of implementation. ACKNOWLEDGMENTS This study was funded partially by the Research Grants Council and the City University of Hong Kong under grant number 7000473. We would also like to thank the Royal Observatory of Hong Kong for their support of this project by providing us the input tide and wind data. REFERENCES [1] Werner, F.E. and Lynch, D.R. (1989), \Harmonic structure of English Channel/Southern Bight tides from a wave equation simulation", Adv. Water Resources, 12, 121-142. [2] Westerink, J.J., Luettich, R.A., Baptista, A.M., Schener, N.W. and Farrar, P. (1992), \Tide and storm surge perdictions using nite element model", J. Hyd. Eng., ASCE, 118(10), 1373-1390. 16

[3] Foreman, M.G.G., Henry, R.F., Walters, R.A. and Ballantyne, V.A. (1993), \A nite element model for tides and resonance along the North coast of British Columbia", J. Geophy. Res., 98(C2), 2509-2531. [4] Henry, R.F. and Walters, R.A. (1993), \Geometrically based, automatic generator for irregular triangular networks", Comm. Num. Meth. Engrg. 9, 555-556. [5] Hardy R.L. (1971), \Multiquadric equations of topography and other irregular surfaces", J. Geophys. Res. 176, 1905-1915. [6] Hardy, R.L. (1975), \Research results in the application of multiquadric equations to surveying and mapping problems", Survey. Mapp., 35, 321-332. [7] Franke R. (1982), \Scattered data interpolation: test of some methods", Math. Comput. 38,181-200. [8] Stead, S. (1984), \Estimation of gradients from scattered data", Rocky Mount. J. math., 14, 265-279. [9] Kansa E.J. (1990), \Multiquadrics - a scattered data approximation scheme with applications to computational uid dynamics - I", Computers Math. Applic. Vol. 19 No. 8/9, 127-145. [10] Kansa E.J. (1990), \Multiquadrics - a scattered data approximation scheme with applications to computational uid dynamics - II", Computers Math. Applic. Vol. 19 No. 8/9, 147-161. [11] Golberg M.A. and Chen C.S. (1994), \On a method of Atkinson for evaluating domain integrals in the boundary element method", Appl. Math. Comput. 60, 125-138. [12] Hon, Y.C. and Mao, X.Z. (1997), \A Multiquadric interpolation method for solving initial value problems", Sci. Comput. 12(1), 51-55. [13] Hon, Y.C. and Mao, X.Z. (1995), \An ecient numerical scheme for Burgers' equations", City Univeristy of Hong Kong, Research Report MA-95-16 and has been accepted to publish at J. Appl. Math. Comp. 17

[14] Hon, Y.C., Lu, M.W., Xue, W.M. and Zhu, Y.M. (1997), \Multiquadric method for the numerical solution of a biphasic mixture model", Appl. Math. Comp. 88(2), 153-175. [15] Golberg, M.A., Chen, C.S. and Karur, S. (1996), \Improved multiquadric approximation for partial dierential equations", Engineering Analysis with Boundary Elements. 18, 9-17. [16] Carlson and Foley T.A. (1991), \The parameter R2 in multiquadric interpolation", Comput. Math. Appl., 21, 29-42. [17] Milroy M.J., Vickers G.W. and Bradley C., \An adaptive radial basis function approach to modeling scattered data", to appear. [18] Gary, W.G. (1982). \Some inadequacies of nite element models as simulators of two-dimensional circulation", Adv. Water Resources. 5, 171-177. [19] Lynch, D.R. and Gary, W.G. (1979). \A wave equation model for nite element tidal computation." Comp. Fluids, 7, 207-228. [20] Micchelli C.A. (1986), \Interpolation of scattering data: distance matrices and conditionally positive de nite functions", Constr. Approx. 2, 11-22. [21] Myers, D.E. (1992), \Kriging, cokriging, radial basis functions and the role of positive de niteness", Comput. Math. Applic. 24(12), 139-149. [22] Fasshauer, G.E. (1996), \Solving partial dierential equations by collocation with radial basis functions", Proceedings of Chamonix, Vanderbilt University Press, Nashville. [23] Tarwater A.E. (1985), \A parameter study of Hardy's multiquadric method for scattered data interpolation", UCRL-54670, Sept. [24] Bogomonly A. (1985), \Fundamental solutions method for elliptic boundary value problems", SIAM J. Numer. Anal., 22, 644-669. [25] Cheng R.S.C. (1987), \Delta-trigonometric and spline methods using the single-layer potential representation", Ph. D. dissertation, University of Maryland. 18

[26] Moridis G.J. and Kansa E.J. (1994), \The Laplace transform multiquadrics method: a highly accurate scheme for the numerical solution of linear partial dierential equations", Appl. Sci. Comp. 1(2). [27] Hong Kong Royal Observatory (1989), \Numerical simulation of shallow water tides in Hong Kong", Tech. Note, 52. Sept. [28] Hon Y.C. (1993), \Typhoon surge in the tolo harbour of Hong Kong - an approach using nite element method with quadrilateral elements and parallel processing technique", Chinese J. Num. Math. Appl., Vol. 15, No. 4, pp. 21-33. [29] Kansa E.J. (1992), \A strictly conservative spatial approximation scheme for the governing engineering and physics equations over irregular regions and inhomogeneously scattered nodes", Computers Math. Applic. Vol. 24, No. 5/6, 169-190.

19

Table 1. Comparison between Modi ed Euler Method and Runge Kutta Method for the case N = 61,
= 20s, and = 4 5 min t

Tide 

MEM RK4

k Exact ? MQ kMax k Exact ? MQ kAve 2.41% 0.28%

0.46% 0.05%

20

R

: d

Current U k UExact ? UMQ kMax k UExact ? UMQ kAve 2.36% 0.52% 0.30% 0.06%

y L

W

x

O

Figure 1 Rectangular basin with 21 X 5 collocation points.

21

8

Average Relative Error ( % )

7

6

5

4

3 N=21 2

1

0 0

N=61 1

2

3

N=41 4

5

6

7 R/d

Figure 2 Average relative errors on computed water surface for N = 21(20)61 and
t = 100s.

22

8

8

Average Relative Error ( % )

7

6

5

4 N=21 3

2

1 N=41 N=61 0 0

1

2

3

4

5

6

7 R/d

Figure 3 Average relative errors on current U for N = 21(20)61 and
t = 100s.

23

8

Average Relative Error ( % ) 6

5

4 Dt=300 3

2 Dt=200

1 Dt=100 Dt=50 0 0

1

2

3

4

Dt=20 5

6

R/d

Figure 4 Average relative errors on computed water surface for N = 41 and dierent
t.

24

Average Relative Error ( % ) 6

5

4

Dt=300

3

2

Dt=200

Dt=100

1

Dt=50 Dt=20 0 0

1

2

3

4

5

6

Figure 5 Average relative errors on current U for N = 41 and dierent
t.

25

R/d

30

25

20 L

E NN

A

15

LO

CH

b

TO

TOLO HARBOUR a

10

Tide Gauge a : Tai Po Kau b : Ko Lau Wan

5

0 0

5

10

15

20

25

30

Figure 6 Sketch of Tolo Harbour of Hong Kong.

26

35

40

45

5.8 5.6 5.4 5.2 5 4.8 4.6 4.4 4.2 4 3.8 2

2.5

3

3.5

4

4.5

Figure 7 Computational domain and MQ collocation points

27

5

18 16 14 12 10 8 6 4 2 0 0

5

10

15

20

Figure 8 Computational domain and triangular elements.

28

25

30

350

300

Tidal level(cm)

−−− actual ___ computed(MQ)

250

:

computed(FEM)

200

150

100

50

0 0

20

40

60

80

100 Time(hour)

120

Figure 9 Measured and computed water surface elevations at Ko Lau Wan during typhoon Gorden (15-21 July, 1989).

29

350

300

Tidal level(cm)

−−− actual ___ computed(MQ)

250

:

computed(FEM)

200

150

100

50

0 0

20

40

60

80

100 Time(hour)

120

Figure 10 Measured and computed water surface elevations at Tai Po Kau during typhoon Gorden (15-21 July, 1989).

30

5.8 5.6

10cm/s

5.4 5.2 5 4.8 4.6 4.4 4.2 4 3.8 2

2.5

3

3.5

4

Figure 11 Flood currents at 26 hrs 20 mins.

31

4.5

5

5.8 5.6

10cm/s

5.4 5.2 5 4.8 4.6 4.4 4.2 4 3.8 2

2.5

3

3.5

4

Figure 12 Ebb currents at 35 hrs 40 mins.

32

4.5

5