Compactly supported radial basis functions for shallow water equations.
This paper presents the application of the compactly supported radial basis functions (CSRBFs) in solving a system of shallow water hydrodynamics equations. The proposed scheme is derived from the idea of piecewise polynomial interpolation using a function of Euclidean distance. The compactly supported basis functions consist of a polynomial which are non-zero on [0,1) and vanish on [1, ). This reduces the original resultant full matrix to a sparse matrix. The operation of the banded matrix system could reduce the ill-conditioning of the resultant coefficient matrix due to the use of the global radial basis functions. To illustrate the computational efficiency and accuracy of the method, the difference between the globally and CSRBF schemes is compared. The resulting banded matrix has shown improvement in both ill-conditioning and computational efficiency. The numerical solutions are verified with the observed data. Excellent agreement is shown between the simulated and the observed data.
Introduction
In marine environments eld monitoring programs are often limited in their ability
to measure spatial and temporal variations in data. This scarcity of information
and a lack of known data about biological and physical phenomenon make
studying these variables extremely di-cult. Because of the complicated inter-action
between hydrological and physical processes mathematical modelling is
commonly used to make predictions of future events. As a consequence, this
1 School of Science and Technology, Open University of Hong Kong, Hong Kong
2 Department of Mathematics, City University of Hong Kong, Hong Kong, Hong Kong
3 2025 University Circle, Las Vegas, Nevada 89119, USA
study aims to numerically simulate the spatial and temporal variation of tidal
currents and water velocities in marine environments.
In this study we introduce an e-cient scheme based on radial basis functions
(RBFs). RBFs were originally devised for scattered geographical data interpolation
by Hardy who introduced a class of functions called multiquadrics [1].
These functions have been widely adopted to solve hyperbolic, parabolic and
elliptic partial dierential equations [2] to [8].
Since the computational cost of using global RBFs is a major factor to
be considered when solving large scale problems with many collocation points,
various techniques have been suggested to improve computational e-ciency. A
common approach is to use domain decomposition. In [9] Dubal used domain
decomposition with a blending technique to cope with the full coe-cient matrix
arising from a multiquadric approximation of the solution of a linear pde. In [10]
Wong et al developed an e-cient multizone decomposition algorithm associated
with multiquadric approximation for a system of non-linear time-dependent
equations. Using this technique the problem of ill-conditioning was reduced
by decreasing the size of the full coe-cient matrix. In addition, multizone
decomposition readily leads itself to parallelization.
However, traditional domain decomposition methods often use iterative corrections
to impose smoothness across internal boundaries and this can increase
computational costs in complicated problems. In this study, we consider an
alternative method to increase computational e-ciency by using compactly
supported RBFs which enable one to work with sparse banded matrices.
The paper is organized as follows. In Section 2 we discuss the basic theory
of the RBF method and the application of compactly supported RBFs is
introduced in Section 3. Section 4 discusses two numerical examples based on
the shallow water equations. The paper ends with a discussion of our results in
Section 5.
2 Application of Radial Basis Functions
The use of radial basis functions to solve PDEs may be viewed as a generalization
of the following multivariate interpolation problem. Let f(x)
a real-valued function and let x
Let be a positive denite radial basis function (see
Denition 3.1) where k x x is the Euclidean distance between x and
Consider an approximation s(x) to f(x)
are the unknown coe-cients to be determined by setting s(x i
This yields the system of linear
@
A
@
@
A
which can be expressed in matrix form as
is an N N
matrices. Since  is positive denite, the matrix A is non-singular so (2.2) has
a unique solution which can be determined by Gaussian elimination.
Generalizing this, if L is a linear partial dierential operator, then an approximation
uN to the solution u of can be obtained by letting
as in (2.1) where f j g are obtained by setting
If the matrix then (2.4) has an unique
solution, and f j g can also be obtained by Gaussian elimination. In general L
is not positive denite, even if  is, so a general theory for this approach is not
yet available. However, numerous numerical studies over the past decade have
shown that L is invertible in many cases and that uN can provide an accurate
approximation to u for su-ciently large N .
In recent years an alternative procedure to solve based on Her-
mite, rather than Lagrange interpolation has been proposed by Fasshauer [11]
and further studied by Franke and Schaback [12, 13]. Here one uses the basis
functions L y ( y), where L y denotes the operation of L with respect to y
in (x y). In this case one can show under quite general conditions that the
corresponding coe-cient matrix is positive denite if  is symmetric, smooth
and positive denite [12]. In this case, one can guarantee the solvability of the
equations (2.4) where  is replaced by L y ( y): However, for many complicated
problems such as the nonlinear hydrodynamic equations studied in this
paper, the Hermite approach may not be feasible as indicated in the paper by
Wong et al [10] and so one must resort to the Lagrange approach in (2.4). Many
of these ideas can be easily generalized to the case where  is only conditionally
positive denite [14] in which one needs to add a polynomial of suitable degree
to the interpolant s(x) in (2.1) and additional conditions must be satised to
guarantee its uniqueness [6].
Although there are many possible radial basis functions, the most commonly
used have been the multiquadrics (MQs)
log r (in R 2 ) and Gaussians These functions
are globally supported and will generate a system of equations in (2.4) with a
full matrix. However, as shown by Madych and Nelson [15], the MQs can be
exponentially convergent so one can often use a relatively small number of basis
elements in which can be computationally e-cient. As a consequence, the MQ
method has been progressively rened and widely used in [3] to [8] for solving
partial dierential equations in various scientic and engineering disciplines.
For example Hon et al [8] used the MQ method to solve a system of time-dependent
hydrodynamics equations and this was extended by Wong et al to
study the coupled hydrodynamics and mass transport equations [16] to predict
pollutant concentrations. Their numerical results demonstrated that the MQ
scheme was more accurate and e-cient than the nite element method. The
MQ method, however requires one to solve a system of equations with a full
matrix. This can be extremely expensive when there are hundreds of collocation
points.
To overcome this problem, we develop a scheme based on compactly supported
radial basis functions (CSRBFs) in this study. Using global RBFs to
solve partial dierential equations can produce a simple algorithm and accurate
results, however solving systems of equations with full matrices can be computationally
expensive and unstable if the matrix is ill-conditioned. CSRBFs can
convert the global scheme into a local one with banded matrices, which makes
the RBF method more feasible for solving large scale problems. The basic
theory of CSRBFs is given in the next section.
Compactly Supported Radial Basis Functions
A family of CSRBFs f l;k g was rst introduced by Wu in [17] and later expanded
by Wendland [19] in the mid 1990s. Generally a CSRBF f l;k (r)g is
expressed in the form
with the following conditions
where p(r) is a prescribed polynomial, r =k x x is the
Euclidean distance and x; x j 2 R d : The index l is the dimension number and
2k is the smoothness of the function.
As discussed previously, to ensure the uniqueness of the interpolant s(x) in
it is su-cient that  be positive denite. We recall the denition of a
positive denite function below:
Denition 3.1 A continuous function  : R d !R is said to be positive denite
on R d , i for all sets of distinct centers
vectors  2 R N nf0g; the quadratic form
is positive. A univariate even function  : R!R is called positive denite on
R d , written as  2 PD d , if the function
is positive denite.
To obtain positive denite CSRBFs Wu used Bochner's theorem[18] which
relates positive denite integrable functions on R d to non-negative Borel measures
on R d . Using this relation Wu proved that an integrable positive denite
compactly supported function  = (r) on R d is continuous if and only if  is
bounded and its d-variate Fourier transforms is non-negative and non-vanishing.
The d-variate Fourier transform of a radial function (r) is given by
f d
where J d 2is the Bessel function of the rst kind of order
given by
There are two approaches to the construction of CSRBFs. They can be constructed
using either the integral (I) or derivative (D) of the function (r) 2
CS \ PD n . The proof technique using the operators D and I in combination
with the d-variate Fourier transform was introduced in [17]. The use of Fourier
transform CSRBFs satises the two recursion formulas
f d
f d
where D, I 2 L 1 (R d ) which are dened by
r
d
dr (r); (3.12)
r
Wu's CSRBFs are obtained using equation (3.12). They are constructed by
starting with a highly smooth univariate positive denite function which are
derived from the following convolution product
of degree (4l + 1) in C 2l \PD 1 . This yields the general form of Wu's compactly
supported functions  l (2r) of degree (4l 2k +1) in C 2l 2k \PD 2k+1 .

Table

some examples of Wu's CSRBFs, where :
represents equality up
to a constant.

Table

1: Examples of Wu's CSRBF functions
It has been shown that the smoothness of Wu's functions deteriorates in
higher dimensional spaces. Wendland [19] has modied Wu's functions using
the criterion in equation (3.13) which are constructed by starting with a low
smoothness truncated power function
l
Applying the integration operator (I) k times on the function  l (r) it will
transform to a polynomial form

Table

some examples of Wendland's functions. We summarize an important theorem
for Wendland's CSRBFs as dened in [19]
Theorem 3.1 For every space dimension d 2 N and every k 2 N there exists
a positive denite compactly supported function  l;k on R d of the form in
equation (3.5) with a univariate polynomial with degree
. The function possesses 2k smooth derivatives around zero
smooth derivatives around 1 and is uniquely determined up to a
constant.

Table

2: Examples of Wendland's CSRBF functions
3;0 (r)
bounds for approximation of f 2 H s (R d ) by CSRBFs are given in [21]
and are of the form
(R d
(R d ) is the Sobolev space with
2.
Recently Fasshauer [11, 22] has given a multilevel approximation algorithm
based on Wendland's functions and demonstrated its behaviour by solving some
simple boundary value problems. We have investigated the amount of work in
the recursive computations and found it excessive for solving non-linear time-dependent
systems. Also, additional errors were generated at each time step.
Hence, this method was not used in this study.
Further explicit formulas for Wendland's functions were given by Fasshauer
[22] for values of 3. The value l is given by b d
the dimension number. The explicit formulae are shown in Table 3.

Table

3: Explicit formula of Wendland's functions
l;0 (r)
The basis function  l;k (r) can be scaled to have compact support on [0; -] by replacing
r with r
The interpolation function dened by equation (2.1)
can be written as
The scaling factor - j can be variable or constant at dierent node points
depending on the nature of the problem. An important unsolved
problem is to nd a method to determine the optimal size of -. In general, the
smaller the value of -, the higher the percentage of zero entries in the coe-cient
matrix. However, this also results in lower accuracy.
To illustrate the algorithm for solving time-dependent problems, we consider
a simple partial dierential equation with time variable t and space variables
in the form
@t
subject to the boundary conditions, for t  0
@x
(0;
@y (x; 0;
The initial condition is
for all (x; y) in [0; m] 2 .
Let data points in [0; m] 2 of which
are interior points; are boundary
points. The numerical discretization for the time derivative of the governing
equation is obtained using a nite dierence method and the spatial derivatives
are approximated by compactly supported radial basis functions. Using forward
dierences the equation (3.18) can be expressed as
where t is the time step. At each time step n, the values u n
at
the interior points are calculated by equation (3.24), while the boundary points
values
are determined by the given boundary conditions.
It is assumed that the values u n are interpolated by
the following function
where r
. The unknown coe-cients  n
are determined
by collocating on the set of distinct points
over the domain
This system of N linear equations may be written in matrix form
where U n and n are single column matrices and Q
is a N N
coe-cient matrix. It is noted that the matrix Q - remains unchanged for a xed
set of data points
and its inverse matrix only needs to be calculated
once. We denote the boundary conditions of the problem in the matrix by
(1) Dirichlet boundary conditions u(x
Neumann boundary @u
boundary conditions @u
@y
These matrices are denoted as
@x
r B+1;N
@x
@x
@x
@y
r C+1;N
@y
@y
7 7: (3.29)
After determining the set of unknown coe-cients f n
j g the numerical values of
the unknown spatial derivatives of u n are approximated by
@x
@x
@y
@y
(3.
It follows that the value of the variable u  A at
1)t at the interior points can be computed by substituting the partial
derivatives (3.32) and (3.33) into the equation (3.24).
In the next section we use the method to solve a real-life model. A system
of linear and non-linear shallow water hydrodynamics equations are considered
separately to compare the accuracy and feasibility of the method with the global
radial basis function method.
Mathematical Models and Numerical Results
We rst apply the proposed method to solve a set of linear shallow water equa-
tions. Another example of a non-linear two-dimensional hydrodynamic model
is considered later to further illustrate the performance of the method. The
numerical results are compared with the results obtained by the global multi-
quadric model in [10]. The programs were written in C++ and all results were
generated using double precision on a Pentium II PC.
4.1 Example 1 - Linear shallow water equation
This simple model allows a comparison of the computed result with the analytical
solution. The equations to be solved are given as
@
@t
@t
where V is a vector of the depth-averaged advective velocities in the x; y directions
is the sea water surface elevation; H is the total depth
of sea level, such that is the mean depth of sea level and g
is the gravitational acceleration. The domain is shown in Figure 1. There
are 205 collocation points in a rectangular channel with length
data points are xed,
of which are interior points; the land boundary points are
and the water-water boundary points are

Figure

1: A rectangular channel with 205 collocation points.
The values at the boundary points are determined by the following given con-
ditions. The land boundary conditions are:
The water-water boundary condition is:
cos !t; at
=s. The initial conditions are:
The analytical solution to this boundary-value problem is given by
cos !t
cos
r
sin
sin !t
cos
The solution corresponds only to the interaction between the incident wave and
the re
ected waves from the wall at Equations (4.34) and (4.35) are
discretized in time using the Taylor method with second order accuracy which
yields
At each time step n, the values of the variables  n and u n at the interior points
are calculated using these two equations. The unknown
spatial derivatives are approximated using a CSRBF  l;k (r) as described in the
previous section. As an illustration, we shall approximate the spatial derivatives
using a simple Wendland's function  -
4;2 (r). Assume that the variables  n and
are approximated using the following positive denite compactly supported
function with a desired scaling factor - j at collocation point
where r
. The unknown coe-cients  n
can
be calculated when all data points
are distinct in the domain
The coe-cient matrices for the N linear equations from (4.47) and (4.48)
can be written in the matrix form@  n
This system involves nding the column vectors
n ]. The coe-cient
matrices
and
R NN are sparse and symmetric. This symmetric
system can be rearranged to form an easy-to-solve banded system by using the
well established LU factorization algorithm.
Having calculated the unknown coe-cients  j and
, the rst partial derivatives
of the function  n
collocation points with
respect to x can be calculated. In the case of  n
j it is obtained by the
following equations
@x
@y
The values of the partial derivatives of the function u n
i can be determined in
a similar manner as the calculation of  n
i . In this simple example, a constant
- can be applied throughout the whole domain. The numerical solutions  n+1
and u n+1
i at the next time step (n + 1)t can be computed subsequently by
substituting the values of the partial derivatives into equations (4.45) and (4.46).

Table

4 shows some results of Wu's and Wendland's functions. The root-
mean-square (RMS) errors of the tidal level () and water velocity (u) in relation
to the analytical solution are compared. The RMS error at three of the interpolation
points and 112 which are situated along the center of the
basin are shown in the table. The RMS error is calculated by
analytical
computed
where T n is the total number of time steps,  analytical
i is the analytical solution
and  computed
i is the simulated result. All results are generated with time step
seconds; the simulation is carried out for 1-hour, i.e. T
Our numerical experiments have shown that the level of accuracy of the
CSRBFs decreases considerably with a decrease in scaling factor as can be compared
to the RMS error in Table 4. The computational results at
are still reasonably accurate with 41% non-zero entries in the sparse matrix.

Table

4: Comparison between results among Wu's and Wendland's CSRBFs
and global multiquadric
tidal level() velocity(u)
Data points(N) 92 102 112 92 102 112
Wendland's Functions
abs
RMS error(cm) 4.78 2.58 4.33 2.51 1.48 2.09
abs
abs
RMS error(cm) 4.99 2.40 1.47 2.25 1.15 2.09
abs
Wu's Functions
abs
RMS error(cm) 2.42 2.89 2.64 2.00 1.39 1.63
abs
Global Multiquadric
abs
The high level of accuracy achieved in this model is due to the simplicity of the
equations and the uniform distribution of the collocation points over the computational
domain. Regarding computational e-ciency, CSRBFs performed
more e-ciently with a saving of about 43% CPU time at
compared to the global MQ scheme.
Wendland's functions are slightly more accurate than that of Wu's functions.
This is shown by the good performance of Wendland's function which generates
a high level of accuracy even with a very sparse matrix, such as the results of
the function  -
4;2 (r) are very close to those of MQ scheme. Our experiments
have shown that CSRBFs with a suitable choice of scaling factor would perform
better than global MQs.
4.2 Example 2 - Non-linear shallow water equations
In this example, we applied the proposed scheme to solve a real-life two dimensional
time-dependent nonlinear hydrodynamics model. Tolo Harbour of
Hong Kong is chosen as the computational domain for comparison purposes.
Tolo Harbour is a semi-enclosed embayment with a very irregular coastline that
surrounds its land boundary; this typical geographical condition makes it well
suited for verifying the proposed methods.
The location of Tolo Harbour is situated approximately
longitude and 22 - 22 0 to 22 - latitude. The embayment of the harbour occupies
an area of 50 km 2 and is long. The width of the embayment varies from
5 km in the inner basin to just over 1 km at the mouth of the channel Harbour.
The tide in Tolo Harbour is a mixed semi-diurnal type with a tidal period of
hours. The overall range of the tidal level is around 0:1 m to 2:7 m. The
measurement of the current
ows is recorded as an average 10 cm=sec in the
channel of the Harbour. The water depth of Tolo Harbour is shallow in Inner
Tolo Harbour less than while the Tolo channel is more than 20 m.
The governing equations are the two-dimensional depth-integrated version
of three dierential equations, namely the continuity equation (4.53) and the
momentum conservation equations (4.54) and (4.55) in the x and y directions
respectively in a
region
These equations are formulated as:
@
@t
@x
@y
@t
@x
@y
@
@x
fvH  a
@t
@x
@y
@
@y
+fuH  a
where u; v are the depth-averaged advective velocities in the x; y directions
is the sea water surface elevation; h is the mean depth of sea
level; H is the total depth of sea level, such that are the
wind velocity components in x; y directions respectively, and W s is the wind
speed given as W
y . C b is the Chezy bed roughness coe-cient;
f is the Coriolis force parameter; g is the gravitational acceleration;  a is the
density of air; w is density of water and C s is the surface friction coe-cient.
The land boundary conditions is dened as
~
where ~
Q represents the velocity vector (u; v), ~n is the direction of the outward
normal vector on the land boundary. The initial conditions are
~
for all x i , y
The time discretization of equations (4.53) to (4.55) is
given by using a forward nite dierence scheme. The resulting nite dierence
equations are written as follows:
@uH
@x
@y
@ n+1
@x
@x
@y
fv  a
Hw
@ n+1
@y
@x
@y
fu  a
Hw
where t is the time step,  n+1 , u n+1 and v n+1 are the iterative solutions at the
1)t at the points . The unknown partial
derivatives are determined by the radial basis function scheme. It is assumed
that the values of the current velocities u n and the surface
elevation  n are approximated using Wendland's function  -
3;1 (r i;j ) at
collocation points
The unknown coe-cients
are determined by collocating with a set
of data points . The computation is performed on the set
of 260 distinct data points on the whole domain, of which 23 data points are on
the water boundary; 107 are interior points and 130 are on the land boundary.
These data points are distributed evenly on the domain as indicated in Figure
2.
The algorithm is slightly dierent compared to the previous examples. At
each time step the numerical solution of the variables on the boundaries are
updated by the following conditions. The open sea boundary condition of the
surface elevation  n dened as
is the input sea surface elevation level on the water boundary.
The input sea surface elevation ^
n on the water boundary at time step n is
estimated using the equation suggested by the Observatory of Hong Kong given
as
where ~
n (t) is the actual tide data measured at a tide gauge; TCOR i is the time
correction parameter and HCOR i
is the tide level correction parameter. The
tide and wind data are the hourly average observed data at two tide gauges
and are obtained from the Observatory of Hong Kong. The location of these
two tide gauges is indicated in Figure 2.
Similarly, the
ow velocities u n on the land boundaries
are updated at each time step using equations (4.68) and (4.69). The current
velocities fu n g and fv n g on the land boundary are obtained from the following
equations
where ~ u n
are the values computed at data point
on the land boundary from the governing equations (4.60) and (4.61),  i is the
outward normal angle at a land boundary point which is computed by taking
the average of the vectors joining the neighbouring points.
To maintain stability, the time-step is chosen to satisfy the following Courant
2.5 3 3.5 4 4.5 54.24.655.45.8Two Major Rivers:
Shing Mun River
a
Inner Tolo
Channel
Tolo Water
Boundary
Two Tide Gauge

Figure

2: Map of Tolo Harbour of Hong Kong showing the locations of Tide
Gauges. The dots () distributed on the map represents the interpolation points
condition
t  d min
gh
where d min is the minimum distance between any two adjacent collocation
points and h is the average water depth between the two points.
The simulation is carried out for 1100 hours from 1 February 1991 to mid
March 1991 with a time step of 30 seconds which means that the total number
of time steps T The performance of the compactly supported RBF
models are compared to the global multiquadric model. The numerical results
are veried with the observed data measured at the tide gauge as shown in

Table

5. It should be noted that functions with higher order of smoothness
can generate slightly more accurate results at the expense of the computational
cost.
It is found that the computations will become signicantly unstable in the
channel Tolo harbour if a constant value of - j is used. Our experiments have
further shown that the accuracy of compactly supported models can be improved
when a relatively larger value of scaling factor - j for the collocation

Table

5: Analysis of the tidal level for the interpolation point at a tide gauge
of the Tolo Harbour computed by global Multiquadric and Wendland's function
RMS absolute Max. CPU time
error (m) error (m) ratio (%)
Global Multiquadric
Wendland's function  3;1 ( r
points on the water boundary is used. It means that the scaling factor - j for
water boundary points is set large enough to cover the whole region in such a
way that the updated information can be propagated in each time step. In this
way, all the entries in the columns of the coe-cient matrix corresponding to
those water boundary points are non-zero. The remaining part of the matrix
can still be banded.
By comparing the RMS errors and absolute maximum errors, the level of
accuracy of the CSRBF model at very close to the global
multiquadric model with a saving of 12% CPU in time. With a smaller
1:210 6 the results of the CSRBF model are still reasonably stable and accurate
throughout the 1.5 months simulation period and it saves 26% CPU time.
The comparison of the simulated tidal level with the observed hourly data
for the period between 23 February 1991 and 27 February 1991 is given in Figure
3. These gures show that there are no signicant dierences in the overall
pattern of the tidal elevation if an appropriate - j is used in the model. The
ooding velocities of the CSRBF model and multiquadric model are compared
in

Figure

4. It shows that the smoothness of CSRBF velocities is similar to
those in the global multiquadric method. Since there is no regular monitoring
of the current velocities in the Tolo Harbour, the prediction of the current velocities
cannot be veried precisely at a single point. However, previous eld
measurement of the current
ow was found to be 10 cm/sec on average in
the channel and has a poor
ushing rate in the inner Tolo harbour, which is
consistent with the numerical predictions.
5 Conclusions and Discussion
A system of mathematical models for the shallow water hydrodynamic system
was constructed to simulate the variation of tidal currents and water velocities.
A compactly supported radial basis function method was employed to approximate
the spatial derivatives of the model. The numerical results of Wu's and
Wendland's CSRBFs were compared with the results of a global multiquadric
scheme. The CSRBF method is computationally e-cient due to the sparse
resultant matrix. Reasonable agreement between the predicted and eld data
was observed.
CSRBF approximation makes RBFs more feasible for solving large-scale
problems. The degree of accuracy is very much dependent on the size of the
local support. The accuracy of the computations can be improved by using a
large scaling factor - j to increase the support for the function. However this
results in a higher computational cost.
The numerical examples have demonstrated an improvement in computational
e-ciency without signicant degradation in accuracy if a suitable scaling
factor is used. For models of complicated systems or irregular domains such as
the model of Tolo Harbour, the numerical accuracy can be improved by using a
variable scaling factor - j to increase the support of the function in partial areas
of the domain on the water boundary or source points.
The eect of the density of data points on the performance of the global
RBF method and the conditioning problem were investigated in our previous
Time in hours (between 23/Feb to 28/Feb 1991)
Tide
Level
Simulated results using one global domain with 260 data points
Observed hourly data at Ko Lau Wan Tide Gauge50150250
Time in hours (between 23/Feb and 28/Feb 1991)
Tide
Level
Simulated results by compact support, scaling factor 1.5km
Observed hourly data at Ko Lau Wan Tide Gauge50150250
Time in hours (between 23/Feb and 28/Feb 1991)
Tide
Level
Simulated results using CSRBF with scaling factor 1.3km
Observed hourly data at Ko Lau Wan Tide Gauge

Figure

3: Comparison of the simulated values of the global MQ and CSRBF
with the observed hourly tide level in Tolo Harbour between 23 Feb and 28 Feb
1991.

Figure

4: Distribution of the
ood velocities in Tolo Harbour at 353 hours
simulating of global MQ and Wendland's CSRBF models
study. We have shown that the condition number of the coe-cient matrix for
the RBF scheme increased rapidly with the increase in the number of data
points see [10].
From a modelling point of view, a more realistic model which takes into
consideration variations along the depth of water in the harbour should be
built. The present model is a relatively simple two-dimensional model where
only the depth-averaged variation is modelled. In marine systems, stratication
often occurs in summer, variation in dierent layers of a water body from the
surface down to the seabed cannot be ignored. These aspects may aect the
prediction accuracy of the model. A more realistic multi-layer model with the
use of radial basis function is currently being investigated.

Acknowledgment

This research is supported by the Research and Development Fund of the Open
University of Hong Kong, # 1.7/96; the Research Grant Committee of City
University of HK, #7000943 and UGC grant. #9040286. The authors wish to
thank the Royal Observatory of Hong Kong for providing meteological data for
numerical computation.



--R


"The theory of radial basis function approximation in 1990"


"The parameter R 2 in multiquadric interpola- tions"
"Improved multiquadric approximation for partial dierential equations"
"Sparse approximation multiquadric in- terpolations"
"A multiquadric solution for shallow water equation"


"Solving dierential equations with radial basis functions: Multilevel methods and smoothing"
"Solving partial dierential equations by collocation using radial basis functions"
"Convergence orders of meshless collocation methods using radial basis functions"
"Generalized Hermite interpolation via matrix-valued conditional positive de nite functions"
"Multivariate interpolation: a variational theory"
"A computational model for monitoring water quality and ecological impacts in marine environments"
"Compactly supported positive de nite radial functions"
"Monotone funktionen, stieltjessche integrale and harmonische analyse"
"Piecewise polynomial, positive de nite and compactly supported radial functions of minimal degree"
"On the smoothness of positive de nite and radial func- tions"
"Error estimates for interpolation by compactly supported radial basis functions of minimal degree"
"On smoothing for multilevel approximation with radial basis functions"
--TR
Generalized Hermite interpolation via matrix-valued conditionally positive definite functions
estimates for interpolation by compactly supported radial basis functions of minimal degree
Solving partial differential equations by collocation using radial basis functions

--CTR
X. Zhou , Y. C. Hon , Jichun Li, Overlapping domain decomposition method by radial basis functions, Applied Numerical Mathematics, v.44 n.1-2, p.241-255, January
hydrodynamic equation;RBF;compact support