Tho spocial case of w eighting íunctions Wt = N' is callcd G alerkin's residual FEM and it is oíten u scd for solving one-dim ensional kincm atic vvave rainfall-runoff m odels.. The rea
Trang 1VNU Journal o f Science, R arth Sciences 24 (2008) 57-65
Stability of spatial interpolation íunctions in íinite element one-dimensional kinematic wave rainfall-runoff models
L u o n g T u a n A n h 1*, R o lf L a r s s o n 2
1 Research Ccntcr for Hydrology and Watcr Rcsourccs,
Institutc o f Hyđro-mctcorological and Environnicntal Sciences
2 Watcr Rcsourccs Enginccring Department, Lund University, Box Ĩ18, S -2 ĨĨ 00 Lund, Sĩoedcn
R eceived 27 M a y 2008; received in revisod fo rm 5 Ju ly 2008
A bstract T h is p a p c r an a ly z e s th c stab ility o f linoar, lu m p e d , q u a d ra tic , a n d cubic spatial
interpolation íunctions in íinitcclcmcnt onc-dimcnsional kỉnematic wavcschcmes for simuỉation of rain fall-ru n o ff proccsscs G a le rk in 's rcsidual m c th o d tra n sfo rm s th c k in c m a tic vvave p artial
diíícrential cquations into a systcm of ordinary diffcrential equations l*hc stability of this system is
analyxed u s in g thc dcíin itio n of the n o rm o f vectors a n d m atrices T h e sta b ility index, o r sing u larity
of tho sy stem , is c o m p u te d b y th c S in g u la r V aluc Decomposition a lg o rith m T h e o scillatio n of th c
solution of the íinite clemcnt onc-đimcnsỉonnl kỉncmatic wave schcmcs rcsults both from thc
sources, a n d fro m th c m u ltip lỉc a tỉo n operator of osciỉlation The re su lts of c o m p u ta tỉo n c x p c rim c n t
and analysis show thc advantagc and disadvantage of diffcrcnt typcs of spatial intcrpoỉntion
functỉons w h c n r i ;.M is a p p lie d fo r rainfall- runoff m o d e lin g b y k in c m a tic w a v c c q u a tio n s.
Keyiuordti: Rainfall-runoff; Kincniatic w ave; Spatial interpolatíon functions; S ingular valuc decom position;
S tability indcx.
1 Introduction
The need for tools w hich havc capability
of sim ulating in ílu en cc of spatial d istrib u tio n
of ram íall and la n d u sc ch an g e on runoff
processes initiated thc d ev clo p m cn t of
hydrodynam ic rainfall-runoff m odols [1, 8]
O ne of the basic a ssu m p tio n s for such m odcls
regards thc cxistcncc of a continuous layer of
vvatcr m oving o v er the w h o lc su ríace of the
catchinents A lthough o b servations sh o w that
such conditions are rare, the assu m p tio n can
” Corresponding autlior Tel.: 84*4*917357025.
&mail: ta n h ễ vkttv.edu.vn
be relaxed by co n sid erin g the total flow to be the result of thc flow from m an y small plots
d raừ ũ n g into a fine netw ork of sm all channels The actual p hysical flow processcs m ay be
q u ite com plicated, bu t for practical ptư p o ses thcrc is n o th in g to be gained írom
in tro d u d n g com ploxity into thc m odels As a com m on vvay of getting op tim al results, thc
o n c-dim cnsional kinem atic w av e m odels [2,
5, 8, 11] are often sclccted T hcse can be
solved by d iffercn t m cth o d s, onc of vvhich is the finite clem cn t m cth o d (FEM) w hich is analyzod in this papor
Tho FEM m o d cls are norm ally dorivcd by the vveighted resid u als m cth o d , vvhich is
57
Trang 258 Luoiỉg Tutrn Anh, Rolf Larssoii / VNU Ịouninl o f Science, Enrth Sciences 24 (2008) 5 7-65
bascd on the principlc th at the solution
residuals should be orthogonal to a set of
vveighting íunctions [7]:
\fì\( h ) ~ f ) W = 0 ,
Q
where:
- 9Ĩ (h) = / : partial differential equation of h;
■ h * Ẹ a A' : estim ated solution;
/
- w, : sot of wGightúìg íunctions;
- N, : functior»s of spatial ordinate;
- fl, : íunctions of từne
A ccording to Pcyrot a n d Taylor Ị9], the
vveighted resid u al m eth o d is a general and
effective tochnique for tran sío rm in g partial
diffcrcntial eq u atio n s (PDE) into systoms of
ordinary differential eq u ations (ODE) Whon
hl,al and N l a re íunctions deíined on a
spatial intorval (clem cnt) the m ethod is called
FEM Tho spocial case of w eighting íunctions
Wt = N' is callcd G alerkin's residual FEM and
it is oíten u scd for solving one-dim ensional
kincm atic vvave rainfall-runoff m odels
The n um erical solutions of tho íinite
elem ent schem os for overland flow and
g ro u n d w a(er flow in onc dim onsional
kincm atic w a v c rainfall-runoff m odcls m ay
oíten ru n into problem s w ith stability and
accuracy d u c to oscillation of tho solution
The schem e m a y be considered stable w hcn
small d istu rb an ce arc not allow cd to grow in
thc num crical proccdure The reasons for
oscillation of th e G alerkin's FEM m cthod for
kincm atic vvave cquations havo boen
discussed by Jaber an d M ohtar [5]
O ne im p o rtan t íactor w hich inAucnccs thc
stability charactoristics of tho m eth o d is tho
choice of sp atial interpolation íunction Jaber
and M ohlar [5] usod linear, lu m p ed and
upvvừid schcm es for spatial approxim ation
and the en h an ccd explicit schem e for
tem poral discretization T hcy a n a ly /c d the
stability of d iííe rc n t schcm es th ro u g h Fourier
analysis and concluded that th e lu m p ed schem e is the m ost suitablo for solution of kinom atic vvave equations
B landíord ct al [2] investigated lincar, quadratic, and cubic Lnterpolation íunctions for sim ulation of one-d im cn sio n al kinem atic vvave by FEM an d ío u n d that q u ad ratic elem onts p ro d u ced th c m o st accuratc solution
w h en tho ừnplicit in teractio n p ro ccd u re vvas usod for tom poral discretixation
T he rosults of th ese researches and tho
m athem atical im plication of G alorkin's FEM shovv that the stability and accuracy of tho íinite clcm ent schcm es does not only d o p en d
on th c typc of spatial in tcrp o latio n íunctions, but also on tho tem poral intogration of tho systom of O D E o ccu rrin g vvhcn FEM is applied for o v erlan d flow kinom atic w av e and groundvvater Boussinesq equations
In the vvorks citcd abovc, th e num erical schcm es h ave bcon bascd on cqui-distant spatial elcm cnts in practical applications, it is often necessary to use elem cn ts of different size, w herc thc d isc rc ti/a tio n reílects iho variation of physical p ro p ertio s of the channol
or th c catchm ents bcing m odclcd The m ain
p u rp o se of this papor is to a n a ly /c tho cffccts
of varying s i/e of spatiai elem ents on the stability of tho solution Furtherm oro, tho origin of instability vvill bo discussed
In tho analysis, thc num crical stability of the various schcm os vvill bo cvaluatcd by invostigating associatcd m atriccs using the Singular V alue D ocom position (SVD) algorithm The íollcnving typos of spatial intcrpolation íu n ctio n s arc invcstigated: linear, lu m p cd , q u ad ratic, and cubic
2 F in ite e le m e n t sch em es for one-
d im e n sio n a l k in e m a tic w ave cq u atio n s
Tho onc-dim ensional kinem atic vvavc
Trang 3Luong Tuan Aìth, Roỉf ỉnrssoìt / V N U Io u rm ỉ ofScieĩĩcc, Enrth Scicĩĩccs 24 (2008) 57-65 59
equations h avc b een used for sừnulation of
the rainfall-runoff proccss in small and
avcrage s i/c river basins w ith stecp slopes
Thoy havc been a p p licd in num crous studies
for hydrological d esig n , ílood íorecasting etc
|2, 3, 6, 8, 11, 12] The one-dim ensional
kincm atic w ave cquations are norm ally
vvritten in tho form of thc continuity equation:
dí dx
and the e q u atio n of m otion for (quasi)
un iíorm flo\v:
vvhoro: h: flow d e p th (m); q : unit-vvidth flovv
(m :/s); r ( x , t ) : eííoctivc rainíall or latcral flow
(m/s); a = s'fí 2 I n ; (i - 5 3: n : M anning
roughness coefficient ( m 1 ' / s ); S’„ : tho suríacc
or bottom slope th at oquals to íriction slopo in
the case of kinom atic w avo approxim ation; x:
spatial co ordinatc (m); and I : tim c (s).
Equations (1) a n d (2) are partial diíícrcntial
equations w hich h av c no goncral analytical
solution Hovvevor, w ith givon initial condition
/?(/={)) and boundary condition numcrical
solutions can bt? found The kinom atic w ave
results from the changcs in flow and since it is
URÌdirectional (from upstream to downstrcam ),
only one b o u n d a ry condition is rcquữed
Principlos of sp atial discretixalion for the
one-dim cnsional kinem atic w av e m odel
using thc FEM m o th o d have been prcscnted
by Ross ct al [11] T h e suríaco area of thc rivcr
basin is d iv id ed in the cross-flow dừ ection
into "strips" Each strip is then divided into
com putational elem onts basod on the
characteristics (e.g slopc) of the basin so that
each elem ent is approxim ately hom ogcncous
For cach co m p utational elem ent, the
variables h(x,t) a n d q(x,t) are approxim ated Ũ1
the form:
h ( x j) * h = ỵ Nị(x)hị(t)-.
q(x.t) e Ậ = z Nịtx)iiị(t)
/=/
(3)
vvhcre: N t(x ): space interpolation íunction
(shapc íunction or vveighting íunction)
It is noted that tho exprcssions (3) should satisíy not only Equation (1) but also the initial condition an d the b o u n d ary condition
The G alcrkin's rosidual m ethod norm alizes thc ap p ro x im ated error vvith shapc íunction ovor the solution dom ain:
ị \ ^ ^ i + ni ỉT L - r i \ ^ ị clx = 0 (4)
The approxim ation (3) com bined w ith the integral (4) transíorm s the partial diííercntial Equation (1) into a system of ord in ary
d iíícrcntial equations, vvhich for oach elem ent (4) takes thc form:
(e) dh
For thc lincar schem e, tho spatial interpolation íunctions can bo dofinod as:
N\(x) = 1 - y , an d N ị ( x ) = y ,
whore y = x / I ; l is tho length of thc elem cnt.
In this case, the m atricos of Equation (5) are w ritten as:
B(e) = Ị
2
A(,) =
- 1 1 -1 1
’ 1_ r
3 6 f(0 _
/ 6
/ 3
/ 2
r(x ,t)
The lu m p ed schem e [5] is bascd on the spatial interpolation íunctions expressed in the forms:
N h = 1 - H
The hcavyside íunction H(x) is d eíin ed as:
H (x)= 0 if X < 0;
H(x) = 1 if x 2 0;
s: distance from no d e j- 1.
( s —— I ' * ( ỉ \
Ị II s - —
Trang 460 l.uoiìg iuau Anh Rolf Larsson / VNU Ịouninl ọ f Science, Larth Sciences 24 (2008) 57-65
The m atriccs for tho lu m p ed schemo of
Equation (5) can bc estim ated in the form:
r
A<e) = —
2
/ 0
0 / The m atrix B(í) and vector f (e) rem ain
the sam c as lin ear schcme
In the casc of q u ad ratic schem e [2], tho
spatial in tcrp o latio n íunctions are:
Nị = l - ĩ y + ĩ y 2:
N 2 = 4 y - 4 y 2;
N ^ - y + l y 2.
T he m atrices for one elem ent are deíined
as followừig:
A (e>
Bw =
2
r(x,t)
For cubic schem c (ono elem ent, íour
nodes), spatial in tcrp o latio n íunctions can bc
exprcsscd in th e íorm s:
/Vj = l - 5 5 y + 9 y 2 - 4 5 /
N 2 = 9 y - 22.5 V2 + 1 3 5 /
N 3 = - 4 5 ^ + 1 8 / - 1 3 5 /
N 4 = y - 4 5 y 2 + 4 5 /
The m atrices for o ne elcm ent are
integrated a n d are p rcsen tcd as:
_8 105 33 560 3 /
140 19
1 6 8 0
/
-/
-33 560
2 7 , 70 27
I
-560 3 140
140 27 560
ụ
70 33
/
-1680 3 /
140 33
5 6 0
560 8 105
B"' =
2
57
’ 80
Ũ)
2_
80
u
■ / ■
3/
8
3/
8 /
— oc
lị.X,1)
For th e vvhole do m ain containing the elem en ts, Equation (5) has the form:
( 6 )
A — + Bq - f = 0
íit
In tho case of using lu m p cd schem e,
m atriccs A; B an d vcctor f for the dom ain
(strip) co n tain in g n elem ents can be presented
in tho íorm s:
tì =
i.ĨL
2 2
0
2 2
0 0 0 « 0 l* - A+ -■
2 2 0
5
0 0
0 0
0 0
0 0
0 0 0 0
1
2
ĩ r*
! úl
Ẵ
1 j 1 l
t
L ỉjl
- C ’ r
Trang 5Luơng Tuan Anh, Rolf ỈẨtrssơn / VNU Ịoum aỉ o/Sciaicc, Earth Sâences 24 (2008) 57-65 61
For overland flovv, the sy stem of ordinary
differontial cq u atio n s (6), can be vvrittcn in
the form:
A — + Bq - C r = 0 ,
vvhere: C: sparse m atrix con tain in g thc size of
elcm cnts; r: vcctor of effectivc rainíall
T he solution of E quation (7) can bc
o b tain ed by v ario u s n um erical m eth o d s, one
of w hich is tho Standard R unge-K utta m ethod
and Successive Linear In tcrp o latio n for
solution of ODE w ith b o u n d aries [4,10]
In o rd cr to a n a ly /e hovv tho stability and
accuracy of thc solution schem es d e p e n d s on
the choice of spatial in tcrp o latio n íunctions,
cq u atio n (7) has been tra n sío rm c d into a
system of linoar algcbraic equations:
vvhore: — = \ : u n k n o w n vector;
ầt
y = C r - Bq : givcn voctor íor explicit
tem poral difforcntial sch em e and estim ated
vcctor for im plicit intoractivo sch em e for oach
ti m e step
3 S ta b ility and erro r a n a ly sis
In o rd er to ev alu ate th e stability of
various íinitc elem en t schem cs, the Singular
Value D ecom position (SVD) alg o rith m will bc
applied It will bc introductíd an d described
bclow togcther vvith the d eíin itio n of somo
cssential vcctor a n d m atrix concepts:
(i) A ccording to tho SVD alg o rith m [4 10],
the m atrix A (raxra) can bc e x p ressed in the
form:
vvhcre u , V: sq u are orth o g o n al m atriccs
(mxm), E : diag o n al m atrix w ith ôn called
singular values of m atrix A
(ii) The norm o f the vcctor X is d efin ed as:
(10)
(iii) The norm of the m atrix A is defined
as thc m axim uin coofficient of extcnsion and can be expressed as:
llA I = II1' s V II < ||U ||||S |||v TII = ||I || = ó'm„ (11)
T he physical im plication of Equation (8) is that onc vector, X, in linear space is transíorm ed
by A into another vector, y This transíorm ation takos three d iííeren t forms: cxtension, com pression, a n d turning
T he stability index, or sin gularity of tho matrix A, can bo d eíined as the ratio of
m axim um extcnsion capacity ovcr the m inim um com prcssion capacity, exprosscd as [4]:
C o n d A ) =
u ỵyTx
min
'max min (12)
vvhere ỔmữS, ổ min: m axim um an d m inim um
singular values of A rospcctivcly
Novv, in o rd cr to stu d y tho stability of the solution schem e, a distu rb an cc (oscillation)
A y is introducod This results in a
co rresponding distu rb an ce (oscillation) A x in
the solution The systcm of linear algcbraic equations (8) w ith and w ithout oscillation bccomcs:
Ax = y = ||y|| < ||A |« ||x|| = 8 max||x|| (13) A(x + Ax) = y + Ay => ||Ay|Ị > 6 min ||Ax||,
vvhere: Ax, ằ y : oscillation vcctor of solution
and oscillation vcctor of crro rs respcctivcly
T his m eans that:
y \ \
(14)
x|| = (xr x ) ,/2
T he relationship (14) show s that the stability of the solution of system (8) d cp en d s
on the stability indcx of the m atrix A w ith a high v alu e of the index indicatừ ig lovver stabiỉity The relationship (14) aiso m eans that the stability index (or singularity of A) m ay
be considered as the m ultip licatio n of
oscillation Ay:
Trang 662 ỉ.uong Tuniì Anh, R olf IẨirsson / VNU Ịoumal o f Science, Lnrth Sciences 24 (2008) 57-65
The u p p er lim it of oscillation (15) can be
estim ated by ap p ly in g the dcíinition of the
norm of vectors an d m atrices:
\\jy\\ = ||CAr - IỈAq\\ <
w herc: ổ : m axim um singular value of
m atrix B; : m axim um singular valuo of
matrix c
Exprossion (16) show s that thc source of
oscillation in clu d e oscillation in the sourcc
tcrm r (effective rainíall) as well as oscillation
in the advcction term accum ulated d u rin g the
com putation process The u p p cr limits of
thosc oscillations d cp en d on the chosen
spatial in terp o latio n íunction, and they are
rclated w ith th e stru ctu re of the m atrices B
and c rcspectivcly T hcsc valuos will bc
com putcd and tho results will bc discussed
belovv for the selectcd types of intorpolation
hm ctions
The solution of tho systcm (8) norm ally
requừ es to invorso m atrix A [5, 12] VVo can
shovv that tho singularity of tho (square)
m atrix A has tho sam e value as thc singularity
of thc inversc m atrix A '7 by using Equation (9):
Application of Singular Value Docomposition
of A'1 gives:
A*1 =U'E'Vt (18)
The decom positions (9) and (18) are
"almost" u n iq u o [10] It m cans that = z ,
and:
C ond(A ) = = C ond(A ') =
ổ—
‘'min max The rclationships (14) an d (19) shovv that
the stability a n d accuracy of solution of
system (8) a re directly relatcd w ith the
singularity of th e h ard m atrix A
4 N u m e ric a l ex p e rim e n ts
In o rd er to vcriíy tho m othodology, so m e basic investigations arc m a d e for d iííe re n t types of in terp o latio n schem es in section 4.1
In section 4.2, the effect of using elem en ts af various lcngths is investigated Pinally, in sectíon 4.3, the inílucnce of different disturbanco sources is analyxcd
4.ĩ Stabiỉity indcx o f matrix A for diffcrcnt typcs ofspatial intcrpoỉationỹunctions
Novv w e assu m e that th e stu d ied strip of
su ríacc area is d iv id ed in to elem en ts of (equal) unit lcngth T he indcx of stability of
m atrix A has been co m p u tcd for various
n u m b crs of elem en ts for oach typc of
in terp o latio n íunction T he rcsults of the
co m p u tatio n s arc preso n tcd in Fig 1
Fig 1 'ITie ch a n g c of sta b iỉity in d cx m atrix A.
T h e num erical ex p erim en ts show that the index of stability is virtually co n stan t for each type of interp o latio n schom c w hen tho
n u m b c r of elcm ents is tw o or highcr It is also clear th at the lu m p c d schem e gives the lovvest value of stability index, w hilc linear,
q u ad ra tic an d cubic schcm cs give 2, 3 and 4 tim es h igher valucs rcspcctively In conclusion, th e lu m p e d schcm e has the
Trang 7Luong Tuuiì A n h , Rtìlf L/irsson t VNU Ịoum al o f Sciciĩce, r.nrth Sáettccs 24 (2008) 57-65 63
highcst order of stability am o n g tho four
studiod num erical schem es
The rcsults of num erical experim ents
prcsonted abovo agreo vvell vvith th e rcsu lts of
analytical Fourior stability an aly sis for
consistent (lincar) and lu m p cd schom cs that
ha vo been presonted in the vvork by Jaber and
M ohtar [5|
4.2 The impact offinite elcnicnt approximations
Num erical exp crim en ts h av e been
conducted in o rd e r to assess tho effect of
olem ent s i/c on stability of the ío u r íinite
olcm ent schem es: linear, lu m p ed , q u ad ratic
and cubic Tho calculations havo b een m ad e
for a strip of 1000 m length, w hich h as boen
approxim ated by tw o elem ents Tho lengths
of the tw o elem en ts h ave been choscn
according to threc diíícrcnt options, with
m ore or loss asym m etric p roportions: option
1 vvith proportìons 1:1, option 2 vvith proportions
1:9, and option 3 vvith p ro p o rtio n s 1:99
The stability index of m atrix A a n d the
m axim um extension capacity of e rro rs of
matrices B and c have boen co m p u tcd and
arc shovvn in Table 1 The results sh o w that
the stability of th e íinibc e lem cn t one-
dim ensional k inem atic vvave schem es does
not only d e p e n d on the ty p c of spatial
interpolation íu n ction, b u t also on the spatial
d iscretừ atio n of th e su ríace strip considered
For all four interp o latio n schem cs, th c lower
the stability is, the m ore d isp ro p o rtio n ate the
elcm ents are At the sam e tim c for all three
options, oach w ith differcnt geom ctric
proportions, the stability is hig h er for lu m p ed
and lincar schcm cs than th at for q u ad ratic
and cubic schem es
A nothcr conclusion is that there are tvvo
m ain causcs for oscillation of the solution
O ne is the oscillation sources, a n d th e other
one is thc m u ltiplication operator
Furtherm ore, it should be pointod o u t that thc cfficiency of thc alg o rith m is an im portant aspect w ith reg ard s to thc choicc of interpolation schcm c for practical applications The linear and lu m p ed schem cs requirc n+1 equations, vvhile q u ad ratic and cubic schem es require 2n+l and 3n+l cq u ations respectively for solving a problem vvith n elem ents
'1'able 1 S tability ind ex of m atrix A
a n d m a x im u m coefficiont of o scillation
C ases of
L u m
-p c d
Q u a d -ratic C u b ic
O p tio n 1 X 5
max 0 8 6 6 0 8 6 6 1.29 1.67
max 404.5 404.5 334.2 198.7
C ond(A ) 3.73 2 0 0 5.83 8.13
O p tio n 2 V' B
max 0 8 6 6 0 8 6 6 1.29 1.67
452.8 452.8 618.5 355.8
C ond(A ) 14.6 1 0 0 41.2 63.1
O p tio n 3 c ti
mox 0 8 6 6 0.866 1.29 1.67
S L 495.0 495.0 680.3 391.3
C ond(A ) 149.6 100.0 448.8 688.6
4.3 The upper limit o f osciỉlation soưrccs for different typcs ofspatial intcrpolation fìuĩctions
ỉí the oscillation o ccu rring at a given tim e
step are su p p o sed to be equal for different typcs of spatial íunctions, th en the u p p c r liirút of source of oscillation will bo related
w ith the m ax im u m sin g u lar values of
m atrices B and c The stru c tu re of thcse
m atrices is d e p en d e d on the type of interpolation íunctions The m axim um singular valuos of B an d c for u n it elem ents
of equal length h ave b een c o m p u ted an d are prescn ted in Table 2
The results show th at for advection oscillation, both thc lincar an d th c lu m p ed schem es give values th at are ncarly
ữ id ep en d en t of the n u m b e r of clcm ents, vvhile the q u ad ratic an d cubic schem es exhibit
Trang 864 Luong Tuan Anh, Rolf Larsso)! / VN U Ịournal o f Science, Enrth Sciences 24 (2008) 57-65
increasing v alu es for increasing nư m ber of
elem ents (sec Fig 2) The cxperim cnt also
shovvs that linear a n d lu m p ed schem es have
the sam e so u rce of oscillation Thoy can also
control thc ad v cctio n oscillation bctter than
q uadratic an d cubic ones Hovvever, the
oscillation of effcctive rainíall com ponent is
bctter controllGd by q u ad ratic and cubic
schem es th an by lu m p ed an d linear ones
T able 2 M ax im u m coefficicnts of sou rce of oscillation
N u rn b e r of
elem cn ts
P a ra
-m c tc rs L in ea r
I.um
-p c d
Q u a d -ratic C ubic
S L 0.500 0.500 0.667 0.375
ổ cnux. 0.809 0.809 0.689 0.398
ó Lno x 0.901 0.901 0.689 0.398
max 0.951 0.951 1.34 1.73
c
5 X &
X c
nu x 0.960 0.960 0.689 0.398
max 0.975 0.975 1.35 1.75
S L 0.971 0.971 0.689 0.398
S L 0.978 0.978 0.689 0.398
Elcmcni*
Fig 2 T he c h a n g e of m a x im u m ex ten sio n capacity
of m atrix B.
5 C o n c lu sio n s
T h is p a p e r analyses the sources an d causes of oscillation of solutions for íinitc elem en t one dim onsional rainfall-runoff
m odcls w h cn different typos of spatia]
in terp o latio n íu n ctio n s is applied for
o v erlan d flow kinom atic w avo sim ulation Lt does so by a p p ly in g tho dcíinition of norm of vectors an d m atricos and the Sirigular Valuo
D ccom position (SVD) algorithm
T ho stru c tu re of m atrix A, w hich contains sizes of the íinitc clem ents, is relatcd to thc type of spatial intorpolation íunction w hich is
ap p licd From thc above proscntod resu lts
an d discussions, it can be concludcd that thc stability indcx or singularity of m atrix A can
be co n sid ered as an eííoct of m ultiplication of oscillation o ccurring d u rin g co m putation
process It will affcct tho stability and
accuracy of thc so lu tio n of íinito elem ent one-
d im cn sio n al k in em atic vvave schem cs, and it
is actually onc of tho m ain causes of oscillation of solutions
T h e rosults of co m p u tatio n experim ont
sh o w thc a d v an tag c and disad v an tag c of
d iffcrcn t typos of spatial intcrpolation
íu n ctio n s w h cn FEM is appliod for rainíall-
ru n o ff k incm atic w av c m odels If thc reason for grovving oscillation is scen as the m ost
im p o rtan t critcrion for assessing stability of num crical schem es, the lu m p ed and lỉncar schom es h av c h ig h er o rd cr of stability th an the q u a d ratic a n d cubic schem es Hovvcver,
w h e n th e lu m p ed schem c is used, tho m atrix
A bccom es a diagonal m atrix and then thc alg o rith m is m ore efficient than all othcr threc typcs of schcm os
T h e rcsults also show that thc íinitc clom ent onc-dim onsional kinom atic vvavc schem es can be im provod by choosing the
m o st suitablo spatial interpolation íunction for d ecreasin g th e singularity of m atrix A and
Trang 9ỈMOìỉg Tiuní A n h , R o ỉ/ ỈJirsson / VNU Ịountaỉ o f Sàcnce, F.nrth Sciences 24 (2008) 57-65 65
m inim i/.e the source of oscillation Tho spatial
interpolation íu n ctions of h ig h er o rd e r do not
alw ays givc im p ro v ed resu lts w h e n íinite
clcm ent m eth o d is used for kincm atic w ave
raĩnfall-runoff m odels
R eíerences
|1] M.B A b b o tt J.c B athurst, J.A Cungo, P.E O '
Connel, J Rasmussen, Structurc of a physically-
bascd distributcd modeling system, / lỉì/droỉ 87
(1986) 61
(2) G.E Blandíord, M.E Meadows, Finitc clcmcnt
simulation of nonlinear kincmatic suríace
runoff / Hydroỉ 119 (1990) 335
[3] V.T Chow, D.R Maidmcnt, L.w Mays, Applied
hydrology, M c Gravv H ill Book Company, 1998.
|4] G.K Porsythe, M.A Malcolm, C.H Moler,
Computer ìĩĩcthod for mathcinatical computations,
Prcnticc-Í lall, N e w jc rs c y , USA, 1977.
[5] 111 Jnber, R.Il Mohtar, Stability and accuracy
of íinitc clcmcnt schcmcs for the
onc-đ im en sio n al kin cm atic w a v e so lu tio n , A d v Water Rcsourcc 25 (2002) 427.
[6] R.s K urothe, N.K Goel, B.s M ath u r, D criv atỉo n
of a curvo n u m b c r an d k in c m a tic vvavc b ase d flo o d írcq u c n cy d istrỉb u tio n , H ydroỉ Sci. / 46 (2001)571
Ị7) C.G Koutitas, Eỉement ợf cơmputatỉonaỉ hi/drnuỉics, Pcntcch Press, London: Plymouth, 1983
[8] L.s K uchm ent, Mathematical modeỉing ợ f rivcr fỉủw (onnuỉation processes, H y d ro m e t Book,
R ussỉa, 1980.
[9] R P eyret, T.D T aylor, Computational methodỉỳ for
fluid fìoio, Springer-Verlag, USA, 1983
[10] w Press, s T eu k o lsk y , vv V etterỉing, B
P lannery, N um ericnl rccipes itĩ ỉ ortrnn, T h e A rt of
S cicntiíic C o m p u tin g , S eco n d cd itio n ,
Cambridgc Univcrsity Press, 1992
[11] B.B Ross# D.N Contractor, v.o Shanholtz,
Pinitc elemcnt modcl of ovcrỉand and channel flow for assessing thc hycỉrologic impact of land usc change, / ìixỊdroỉ 41 (1979) 11.
[12] Y Y uyam a, Rcgionaỉ draiìtage nnnlỉ/sis bìỊ nìathciĩuitical ìtiodeỉ simuỉatiữH, N ational Research
In stitu tc of A griculturnl Hnginoering, Japan, 19%.