VNU Journal o f Science, Rarth 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 1Research Ccntcr for Hydrology and Watcr Rcsourccs, Institutc o f Hyđro-mctcorological and Environnicntal Sciences Watcr Rcsourccs Enginccring Department, Lund University, Box Ĩ18, S-2ĨĨ 00 Lund, Sĩoedcn R eceived 27 M ay 2008; received in revisod form July 2008 A bstract T h is p a p c r analy zes th c stability o f linoar, lu m p e d , q u ad ratic, a n d cubic spatial interpolation íunctions in íinitcclcmcnt onc-dimcnsional kỉnematic wavcschcmes for simuỉation of rainfall-ru n o ff proccsscs G a le rk in 's rcsidual m cth o d tran sfo rm s th c k in cm atic vvave partial diíícrential cquations into a systcm of ordinary diffcrential equations l*hc stability of this system is analyxed u sin g thc dcíinition of the n o rm of vectors a n d m atrices T h e stability index, or singularity Decomposition a lg o rith m T h e oscillation of thc solution of the íinite clemcnt onc-đimcnsỉonnl kỉncmatic wave schcmcs rcsults both from thc sources, an d fro m th c m u ltip lỉcatỉo n operator of osciỉlation The re su lts of c o m p u ta tỉo n cx p crim cn t and analysis show thc advantagc and disadvantage of diffcrcnt typcs of spatial intcrpoỉntion of tho system , is co m p u te d by th c S in g u la r V aluc functỉons w h cn r i ;.M is ap p lie d for rainfall- runoff m o d elin g b y kin cm atic w a v c cquations Keyiuordti: Rainfall-runoff; Kincniatic wave; Spatial interpolatíon functions; Singular valuc decomposition; Stability indcx Introduction be relaxed by considering the total flow to be the result of thc flow from m any small plots draừũng into a fine netw ork of sm all channels The actual physical flow processcs m ay be quite com plicated, but for practical ptư poses thcrc is nothing 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 optim al results, thc onc-dim cnsional kinem atic w ave m odels [2, 5, 8, 11] are often sclccted Thcse can be solved by differcnt m cthods, onc of vvhich is the finite clem cnt m cthod (FEM) w hich is analyzod in this papor The need for tools w hich havc capability of sim ulating iníluencc of spatial d istribution of ram íall and lan d u sc change on runoff processes initiated thc d evclopm cnt of hydrodynam ic rainfall-runoff m odols [1, 8] O ne of the basic assum ptions for such m odcls regards thc cxistcncc of a continuous layer of vvatcr m oving over the w holc suríace of the catchinents A lthough observations show that such conditions are rare, the assum ption can Tho FEM m odcls are norm ally dorivcd by the vveighted residuals m cthod, vvhich is ” Corresponding autlior Tel.: 84*4*917357025 &mail: ta n h ễ vkttv.edu.vn 57 58 Luoiỉg Tutrn Anh, Rolf Larssoii / VNU Ịouninl o f Science, Enrth Sciences 24 (2008) 57-65 bascd on the principlc that the solution residuals should be orthogonal to a set of vveighting íunctions [7]: \fì\(h )~ f ) W = , 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 an d Taylor Ị9], the vveighted residual m ethod is a general and effective tochnique for transíorm ing partial diffcrcntial equatio n s (PDE) into systoms of ordinary differential equations (ODE) Whon hl,al and N l are 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 uscd for solving one-dim ensional kincm atic vvave rainfall-runoff m odels The num erical solutions of tho íinite elem ent schem os for overland flow and groundw a(er flow in onc dim onsional kincm atic w av c rainfall-runoff m odcls may oíten ru n into problem s w ith stability and accuracy d u c to oscillation of tho solution The schem e m ay be considered stable w hcn small distu rb an ce arc not allow cd to grow in thc num crical proccdure The reasons for oscillation of the G alerkin's FEM m cthod for kincm atic vvave cquations havo boen discussed by Jaber and M ohtar [5] O ne im p o rtan t íactor w hich inAucnccs thc stability charactoristics of tho m ethod is tho choice of spatial interpolation íunction Jaber and M ohlar [5] usod linear, lum p ed and upvvừid schcm es for spatial approxim ation and the en h an ccd explicit schem e for tem poral discretization Thcy an aly /cd the stability of d iííercn t schcm es through Fourier analysis and concluded that the lum ped 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-dim cnsional kinem atic vvave by FEM and íound that q u adratic elem onts produced thc m ost accuratc solution w hen tho ừnplicit interaction proccdure vvas usod for tom poral discretixation T he rosults of these 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 dopend on thc typc of spatial intcrpolation íunctions, but also on tho tem poral intogration of tho systom of ODE occurring vvhcn FEM is applied for overland flow kinom atic w ave and groundvvater Boussinesq equations In the vvorks citcd abovc, the num erical schcm es have bcon bascd on cqui-distant spatial elcm cnts in practical applications, it is often necessary to use elem cnts of different size, w herc thc discrcti/atio n reílects iho variation of physical propertios of the channol or thc catchm ents bcing m odclcd The m ain purpose 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 íunctions arc invcstigated: linear, lum pcd, quadratic, and cubic F inite elem en t schem es for oned im en sio n al k in em atic w ave cquations Tho onc-dim ensional kinem atic vvavc Luong Tuan Aìth, Roỉf ỉnrssoìt / V N U Iourm ỉ ofScieĩĩcc, Enrth Scicĩĩccs 24 (2008) 57-65 equations havc been used for sừnulation of the rainfall-runoff proccss in small and avcrage s i/c river basins w ith stecp slopes Thoy havc been applicd in num crous studies for hydrological design, 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: ơh õq nv * +T~ = '■ 15 Bw = for one elem ent are deíined / 15 21 30 15 15 2 1' • 2 2_ 80 57 80 ■/ ■ 00 ” ĨÕ 81 80 nư 8] 80 ĩõ Ũ) 3/ lị.X,1) 3/ / — For the vvhole dom ain containing the elem ents, Equation (5) has the form: A — + Bq - f = (6) íit In tho case of using lum pcd scheme, m atriccs A; B and vcctor f for the dom ain (strip) containing n elem ents can be presented in tho íorm s: / 30 / 15 8/ 15 / 57 ’ 80 80 ~7Õ 57 80 — oc The m atrix B(í) and vector f (e) rem ain the sam c as linear schcme In the casc of quadratic schem e [2], tho spatial intcrpolation íunctions are: B"' = 57 80 nu The m atriccs for tho lum ped schemo of Equation (5) can bc estim ated in the form: r / A ||Ay|Ị > 6min ||Ax||, vvhere: Ax, ằ y : oscillation vcctor of solution and oscillation vcctor of crrors respcctivcly This m eans that: (14) y\\ T he relationship (14) show s that the stability of the solution of system (8) dcpends on the stability indcx of the m atrix A w ith a high value of the index indicatừig lovver stabiỉity The relationship (14) aiso m eans that the stability index (or singularity of A) m ay be considered as the m ultiplication of oscillation Ay: ỉ.uong Tuniì Anh, Rolf IẨirsson / VNU Ịoumal o f Science, Lnrth Sciences 24 (2008) 57-65 62 Ay = CAr - B A q (15) The u p p er lim it of oscillation (15) can be estim ated by applying the dcíinition of the norm of vectors and matrices: \\jy\\ = ||CAr - IỈAq\\ < < ||c4r[+||Bâí | s +S I M (16) ổ : m axim um singular value of matrix B; : m axim um singular valuo of wherc: N u m erical experim ents In order to vcriíy tho m othodology, som e basic investigations arc m ad e for diííeren t types of interpolation schem es in section 4.1 In section 4.2, the effect of using elem ents af various lcngths is investigated Pinally, in sectíon 4.3, the inílucnce of different disturbanco sources is analyxcd matrix c Exprossion (16) show s that thc source of oscillation include oscillation in the sourcc tcrm r (effective rainíall) as well as oscillation in the advcction term accum ulated during the com putation process The u ppcr limits of thosc oscillations d cp en d on the chosen spatial interpolation íunction, and they are rclated w ith the structure of the m atrices B and c rcspectivcly Thcsc valuos will bc com putcd and tho results will bc discussed belovv for the selectcd types of intorpolation hmctions The solution of tho systcm (8) normally requừes 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): 4.ĩ Stabiỉity indcx of matrix A for diffcrcnt typcs ofspatial intcrpoỉationỹunctions Novv w e assum e that the studied strip of suríacc area is divided into elem ents of (equal) unit lcngth T he indcx of stability of m atrix A has been com putcd for various num bcrs of elem ents for oach typc of interpolation íunction T he rcsults of the com putations arc presontcd in Fig A~' = V E 'l ỉ r (17) Application of Singular Value Docomposition of A'1gives: A*1=U'E'Vt (18) The decom positions (9) and "almost" uniquo [10] It m cans that and: C ond(A ) = ổ— (18) are =z , = C ond(A ') = ‘'min max The rclationships (14) and (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 hard m atrix A Fig 'ITie ch an g c of stab iỉity indcx m atrix A T he num erical experim ents show that the index of stability is virtually constant for each type of interpolation schom c w hen tho n um bcr of elcm ents is tw o or highcr It is also clear that the lu m p cd schem e gives the lovvest value of stability index, w hilc linear, q u ad ratic and cubic schcm cs give 2, and tim es higher valucs rcspcctively In conclusion, the lu m p ed schcm e has the Luong Tuuiì A nh, Rtìlf L/irsson t VNU Ịoum al o f Sciciĩce, r.nrth Sáettccs 24 (2008) 57-65 highcst order of stability am ong tho four studiod num erical schemes The rcsults of num erical experim ents prcsonted abovo agreo vvell vvith the rcsults of analytical Fourior stability analysis for consistent (lincar) and lum pcd schom cs that vo been presonted in the vvork by Jaber and M ohtar [5| 4.2 The impact offinite elcnicnt approximations Num erical expcrim ents have been conducted in o rd er to assess tho effect of olem ent si/c on stability of the ío u r íinite olcm ent schemes: linear, lum ped, q u adratic and cubic Tho calculations havo been m ade for a strip of 1000 m length, w hich has boen approxim ated by tw o elem ents Tho lengths of the two elem ents have been choscn according to threc diíícrcnt options, with m ore or loss asym m etric proportions: option vvith proportìons 1:1, option vvith proportions 1:9, and option vvith proportions 1:99 The stability index of m atrix A an d the m axim um extension capacity of erro rs of matrices B and have boen com putcd and arc shovvn in Table The results show that the stability of the íinibc elem cn t onedim ensional kinem atic vvave schem es does not only d ep en d on the typc of spatial interpolation íunction, b u t also on the spatial discretừation of th e su ríace strip considered For all four interpolation 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 higher for lum ped and lincar schcm cs than that for q u adratic and cubic schemes Anothcr conclusion is that there are tvvo main causcs for oscillation of the solution One is the oscillation sources, an d th e other one is thc m ultiplication operator c 63 Furtherm ore, it should be pointod out that thc cfficiency of thc algorithm is an im portant aspect w ith regards to thc choicc of interpolation schcmc for practical applications The linear and lum ped schem cs requirc n+1 equations, vvhile quadratic and cubic schem es require 2n+l and 3n+l cquations respectively for solving a problem vvith n elem ents '1'able S tability index of m atrix A and m axim um coefficiont of oscillation C ases of stu d y O ption X5 max max O p tio n Linear L um p cd 6 6 Q uadC ubic ratic 1.67 1.29 404.5 404.5 334.2 198.7 0 5.83 8.13 6 1.29 1.67 452.8 618.5 355.8 C ond(A ) 3.73 V' B 6 max 452.8 O p tio n C ond(A ) 14.6 c ti 6 0 41.2 63.1 0.866 1.29 1.67 495.0 495.0 680.3 391.3 C ond(A ) 149.6 100.0 448.8 688.6 mox S L 4.3 The upper limit of osciỉlation soưrccs for different typcs ofspatial intcrpolation fìuĩctions ỉí the oscillation occurring at a given tim e step are supposed to be equal for different typcs of spatial íunctions, then the up p cr liirút of source of oscillation will bo related w ith the m axim um singular values of matrices B and The structure of thcse matrices is d ep en d ed on the type of interpolation íunctions The m axim um singular valuos of B and for unit elem ents of equal length have been com puted and are prescnted in Table The results show that for advection oscillation, both thc lincar and thc lum ped schem es give values that are ncarly ữ idependent of the n u m b er of clcm ents, vvhile the quadratic and cubic schem es exhibit c c Luong Tuan Anh, Rolf Larsso)! / VNU Ịournal of Science, Enrth Sciences 24 (2008) 57-65 64 increasing values for increasing nưm ber of elem ents (sec Fig 2) The cxperim cnt also shovvs that linear an d lum ped schem es have the sam e source of oscillation Thoy can also control thc advcction oscillation bctter than quadratic and cubic ones Hovvever, the oscillation of effcctive rainíall com ponent is bctter controllGd by q uadratic and cubic schem es than by lu m p ed and linear ones C o n clu sio n s N u rn b er of P ara elem cnts m c tc rs L inear I.um p cd Q u ad - 1.0 1.0 1.16 1.55 S L 0.500 0.500 0.667 0.375 S L 0.866 0.866 1.29 1.67 ổ nux c 0.809 0.809 0.689 0.398 XB 1.0 1.0 1.33 1.71 ó no Lx 0.901 0.901 0.689 0.398 T his paper analyses the sources and causes of oscillation of solutions for íinitc elem ent one dim onsional rainfall-runoff m odcls w hcn different typos of spatia] interpolation íunctions is applied for overland flow kinom atic w avo sim ulation Lt does so by ap p ly in g tho dcíinition of norm of vectors and m atricos and the Sirigular Valuo D ccom position (SVD) algorithm T ho stru ctu 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 applicd From thc above proscntod results and discussions, it can be concludcd that thc stability indcx or singularity of m atrix A can be considered as an eííoct of m ultiplication of oscillation occurring during com putation 0.951 0.951 1.34 1.73 process It will affcct tho stability and 0.940 0.940 0.689 0.398 X& * niax 1.0 1.0 1.35 1.74 Xc 0.960 0.960 0.689 0.398 X * 0.975 0.975 1.35 1.75 S L 0.971 0.971 0.689 0.398 S L 1.0 1.0 1.35 1.75 SL 0.978 0.978 0.689 0.398 T able M ax im u m coefficicnts of source of oscillation X m *x ^ max o nitx c nu x max ratic C ubic Elcmcni* Fig T he c h a n g e of m a x im u m extension capacity of m atrix B accuracy of thc solution of íinito elem ent onedim cnsional kinem atic vvave schemcs, and it is actually onc of tho m ain causes of oscillation of solutions T h e rosults of com putation experim ont show thc ad v an tag c and disadvantagc of diffcrcnt typos of spatial intcrpolation íunctions w hcn FEM is appliod for rainíallrunoff kincm atic w avc 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 lum ped and lỉncar schom es havc higher o rdcr of stability than the q u ad ratic an d cubic schemes Hovvcver, w hen the lum ped schem c is used, tho m atrix A bccom es a diagonal m atrix and then thc algorithm 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 kinomatic vvavc schem es can be im provod by choosing the m ost suitablo spatial interpolation íunction for decreasing the singularity of matrix A and ỈMỉg Tiuní A nh, R o ỉ/ ỈJirsson / VNU Ịountaỉ o f Sàcnce, F.nrth Sciences 24 (2008) 57-65 65 minimi/.e the source of oscillation Tho spatial interpolation íunctions of higher o rd er not always givc im proved results w h en íinite clcm ent m ethod is used for kincm atic w ave raĩnfall-runoff m odels [6] R eíerences [8] L.s K uchm ent, Mathematical modeỉing ợ f rivcr đ im en sio n al kincm atic w av e Water Rcsourcc 25 (2002) 427 R.s K urothe, N.K Goel, B.s M athur, D crivatỉon of a curvo n u m b cr and k in cm atic vvavc b ased flood írcqucncy 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 fỉủw J.c B athurst, J.A Cungo, P.E O ' Connel, J Rasmussen, Structurc of a physicallybascd 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, Mc Gravv H ill Book Company, 1998 |4] G.K Porsythe, M.A Malcolm, C.H Moler, |1] M.B A b b o tt 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- A dv solution, (onnuỉation processes, H y d ro m e t Book, R ussỉa, 1980 [9] R Peyret, T.D T aylor, Computational methodỉỳ for fluid fìoio, Springer-Verlag, USA, 1983 [10] w Press, s T eukolsky, vv V etterỉing, B Plannery, Num ericnl rccipes itĩ ỉ ortrnn, T h e A rt of Scicntiíic [11] C o m p u tin g , S econd cd ition, Cambridgc Univcrsity Press, 1992 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 Institutc of A griculturnl Hnginoering, Japan, 19% ... bo cvaluatcd by invostigating associatcd m atriccs using the Singular V alue D ocom position (SVD) algorithm The íollcnving typos of spatial intcrpolation íunctions arc invcstigated: linear,... thc singularity of thc inversc m atrix A '7 by using Equation (9): 4.ĩ Stabiỉity indcx of matrix A for diffcrcnt typcs ofspatial intcrpoỉationỹunctions Novv w e assum e that the studied strip of. .. solution of system (8) dcpends on the stability indcx of the m atrix A w ith a high value of the index indicatừig lovver stabiỉity The relationship (14) aiso m eans that the stability index (or singularity