Parameter Selections in Simulating the Physical Diffusion Phenomena of Suspended Load by Low Order Differential Scheme Numerical Dispersion

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1 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach 013 Paamete Selectios i Simlatig the Physical Diffsio Pheomea of Sspeded Load by Low Ode Diffeetial Scheme Nmeical Dispesio Shai Che *1, Jig Wag State Key Laboatoy of Wate Resoces ad Hydopowe Egieeig Sciece, Wha Uivesity Wha, Hbei, Chia *1 liveagela@wh.ed.c; jwag@wh.ed.c Abstact I this pape, the pocess of the itodctio of meical diffsio is ivestigated whe low ode diffeetial scheme i covectio diffsio eqatio fo discete sspeded load is i se. Accodig to the eslts of istaces calclatio, the optimal choice fo paametes whe physical diffsio is simlated by meical diffsio is obtaied. Fially, compaiso betwee seveal commoly sed low ode schemes is codcted ad the most viable low ode scheme i simlatig physical diffsio by meical diffsio is poposed i dealig with sspeded load. Keywods Diffeetial Scheme; Covectio Diffsio Eqatio of Sspeded Load; Nmeical Diffsio Itodctio I geeal, sig the low ode scheme fo meical calclatios will moe o less itodce meical diffsio (Xie 1990). Diffeet schemes have vayig degees of meical diffsio. I theoy this is ot codcive to the eact soltio of meical calclatio, bt tilizig the meical diffsio, sch as to simlate the physical diffsio pheomeo, is also a effective method (Yag 1993). As fo how to se the meical diffsio to simlate the physical diffsio, ad how to select paametes to achieve a acceptable level of the simlatio eslts, they ae the isses to be discssed i this pape. Covectio-Diffsio Eqatio fo Oe- Dimesioal Sspeded Load Eqatio fo sspeded load movemet is (Zheg ad Zhao 001): AS QS S A AD S t t ' S ( ) Itodcig the flow cotiity eqatio, the eqatio above ca be ewitte as: S Q S S S A t A A A t ' 1 ( ) S AD whee D the logitdial diffsio coefficiet, fo sedimet diffsio, it ca be appoimated as D 0.5 h * Geeally, fo taspotatio isse of sspeded load, the logitdial diffsio of sspeded load is mch smalle tha the logitdial covictio of sedimet, which is ofte ovelooked. If the caie velocity ad diffsio coefficiet D ae costats withot cosideig the soce item, the covectio diffsio eqatio will be t S S D S Fo this eqatio, the mai meical difficlty is to calclate the covectio tem becase it stictly demads the cosevatio of matte, while the diffeetial method fo meical soltio ofte caot achieve this. If the soltio to covectiodiffsio is cosideed de the pemise of a good soltio to covectio diffsio tem, the pobability of sccessfl meical soltio wold be highe. If igoig the diffsio tem, we ca get the pe covectio eqatio S S t 0 1

2 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach Establishmet of Diffeetial Scheme Select Upwid scheme discete pe covectio eqatio, whe 0, the diffeetial eqatio is (Coat 198) S S C ( S S ) 1 j j j j1 whee C is Cot mbe ad C C t/ The diffeetial eqatio ca be edced to 1 Sj Sj Sj Sj 1 C 0 t The fist item i the above eqatio is the fowad diffeece qotiet of S / t, ad the secod tem is backwad diffeece qotiet of S / t, the C ca be daw. We will eam the compatibility, stability ad covegece of the scheme as follows. 1 Epad S ad S j 1 o poit (j,) j 1 S 1 S Sj S j ( ) t t ( )... jt t t S 1 S S S j1 j ( ) j ( )... j Take them ito the diffeetial eqatio, ad se the pe covectio eqatio to get S S C(1 C) S t t Assme D The C (1 C) t t S S D S Compaig to the pe covectio eqatio, oly the ight tem of the eqatio teds to be zeo whe the space ad time step ae small eogh, which meas they ae compatible. Takig t C / C ito D, we ca get C(1 C ) D whee C, 0 1, whe 0, t 0, ad C D 0. So the diffeece eqatios ad diffeetial eqatios ae compatible, as the stability coditio fo Upwid is t / 1. Usig La eqivalece theoem agai we ca get that the soltio of diffeetial eqatio coveges to the soltio of diffeetial eqatios. Fom the above poof it is clea that, Upwid scheme dispeses a pe covectio eqatio ad a eqatio simila to covective diffsio ca be obtaied. Althogh the diffeetiated items D S / ca make the eqatio compatible, calclatio shows that the item also makes the soltio of diffeetial eqatios o loge covege to the soltio of the oigial diffeetial eqatio. Istead it coveges to the soltio of the covectio diffsio eqatio, which meas the implicit meical diffsio, esltig fom the limitatio of, t, which caot be ifiitely close to zeo. Howeve, sice it is kow that diffeece eqatios covege to the covectiodiffsio eqatio, the it is sed to idetify the eqied paamete vales, so that the diffeetial eqatios ca covege pecisely to diffeetial eqatios, amely meical diffsio ca simlate the physical diffsio well. The Istace Aalysis of Paametes Effect As diffeetial eqatio is simila i fom with covectio diffsio eqatio, D might be called meical diffsio coefficiet. It is clea that the closeess betwee meical diffsio coefficiet D ad physical diffsio coefficiet D ca detemie the accacy of the meical diffsio i simlatig physical diffsio, ths the poblem is to fid the ight paametes, so that D ca be close to D (Papadakis ad Metaas 011). O the othe had, thee paametes ca detemie D which ae velocity, space iteval ad time step t, whee flow ate shold be i accodace with the actal egieeig vale, ad is sbject to the estictios o the boday simlatio accacy (it will take o diffeet vales i accodace with the diffeet size of the poject ad the compleity of the bode. Usally, fo eample, the smalle the size of is, the moe pecise the boday simlatio eslts will be. Bt it will icease the amot of calclatio), oly the time step t has wide choice. By selectig diffeet t we ca get the coespodig meical diffsio coefficiet D. The compaig D with diffeet logitdial diffsio coefficiet D, the impact of t o calclated eslts ca be eamied (Gasioowski 013). Bild the followig model: 13

3 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach 013 Model 1 Legth is 10km, width is m, wate depth is 4m, Chezy coefficiet C is 5, ad flow ate is 0.5m/s, =00m. Take espectively t =400s, 399.5s, 399s, 360s, 300s, 00s The calclatio eslts ae show as follows (Table 1). t TABLE 1 CALCULATION RESULTS (U=0.5M/S) Paametes C D D I the above table, oly whe t =399.5s, the diffeece betwee meical diffsio coefficiet D ad logitdial diffsio coefficiet D is the smallest at At the same time, the diffeece betwee C ad 1 is , whe t deceases by 0.5s to 399s. The two diffsio coefficiets diffes i a ode of magitde, idicatig that oly whe C is vey close to 1 the physical diffsio ca be simlated moe accately. O the othe had, the meical diffsio coefficiet D will icease with the decease of C. Whe C teds 0, a soltio with lage eo will be got. Hee the time step is accate to the etet of 0.5, which is ot codcive to calclatio. Sice the time step may be ifleced by the flow velocity, the flow ate is chaged ad the model is modified as follows: Model Flow velocity =1m/s, takig t =400s, 300 s 00s, 199.5s, 199s, 198s othe paametes ae chaged. I the eslts (Table ) whe t is geate tha, meical diffsio coefficiet is egative. It will epeat the last calclatio law whe they ae eqal. I fact if the fomla is chaged fo cotadictios. D, we ca fid the C(1 C) t D t Whe =1m/s, D ( t)/, obviosly eeds to be lage tha t to make D positive, i fact, this is cased by the faile to satisfy the stability coditio t / 1. Ad, fom aothe pespective, the accacy of D shows diect liea elatioship with the pecisio of as well as t, which demads highly fo ad t, i the actal calclatio, it is ofte impossible to meet sch a eqest, so othe flow ates shold be take to calclate agai. t TABLE CALCULATION RESULTS (U=1M/S) Paametes C D D Model Flow velocity =0.8m/s, takig t =50s, 49.5s, 49s, 00s, 100s, 50s The emaiig paametes ae chaged. Model 4 Flow ate =0.m/s, takig t =1000s, 999.5s, 999s, 998s, 900s, 800s The emaiig paametes ae chaged. Calclatio eslts ae show i Table 3 ad Table 4 espectively. 14

4 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach t TABLE 3 CALCULATION RESULTS (U=0.8M/S) Paametes C D D =0.5m/s; Whe is i the iteval (0,1], the simlatio is bette whe the time step vale is 0.5s smalle tha the ʺ t that makes C =1, bt the diffeece of 0.5s is ot ideal; Whe is i the iteval (0,1], althogh the close C is to 1, the bette the simlatio is, howeve withi a small age close to 1, it is ot i lie with this le. As show i the followig table. TABLE 5 VARIATIONS IN PARAMETERS WITH DIFFERENT U (m/s) t Paametes C D D D D % TABLE 4 CALCULATION RESULTS (U=0.M/S) t Paametes C D D Combiig the above fo models; we ca daw the followig les: Whe is i the iteval (0,1], the logitdial diffsio coefficiet D iceases as icease, ad the maimm vale is close to 0.16 (this is the maimm vale whe the wate depth is 4m ad Chezy coefficiet is 5, depedig o the cicmstaces, the vale will be diffeet); Whe is i the iteval (0,1], if is small, the meical diffsio simlatio of physical diffsio is bette, which is the closest whe C D =0.m/s =0.5m/s =0.8m/s =1.0m/s lgδt FIG. 1 DISTRIBUTION OF C WITH GROWTH OF THE TIME STEP t (IN LOG FORM) =0.m/s =0.5m/s =0.8m/s =1.0m/s lgδt FIG. DISTRIBUTION OF D WITH GROWTH OF THE TIME STEP t (IN LOG FORM) Obviosly, i the descedig pocess of, C is gettig close to 1, bt D D is the miimm whe =0.5m/s. The eistece of a smalle diffeece ca be 15

5 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach 013 tested by chagig the vale of, howeve it is o loge stdied hee. I fact, i ode to get the paametes coditios of meical diffsio that accately simlate physical diffsio, fomlas ae sed to easo evesely, assme D D We get: D t D 0.5* h 0.5 g h C Simplify it to be: 0.5 gh t C By assmig 0.5 gh k, we ca obtai: C k t Obviosly, t is a liea fctio of, whee k is a costat elated to the paametes of the physical model. Take =0.5, h =4, C =5, =00m ito the fomla the the calclatio shows t = which is vey close to the above 399.5, so the diffeece betwee the two diffsio coefficiets is isigificat. Bt this step size is vey favoable to calclatio; as it is tedecy to take a itege as mch as possible. Althogh i diffeet egieeig cases, diffeet paametes of the physical model ca impove the vale of t, its vale depeds completely o the diffeetial scheme itself. I ode to keep the geeality, the followig will hoizotally cotast a vaiety of commoly sed diffeetial schemes de the cet model paamete coditios to eamie whethe a scheme ca achieve a elatively satisfyig step vale. Compaiso amog Vaios Diffeece Schemes Althogh vaios diffeetial schemes have diffeet compatibility ad stability coditios, as log as etactig meical diffsio coefficiet afte dispesig pe covectio eqatios espectively, the eqivalet ify with the physical diffsio coefficiet, we ca obtai the calclatio fomla fo step size (Table 6). Obviosly, these thee schemes of Dobbis, Upwid ad implicit have the same method to select the step size, ad they also have the same chaacteistics epeseted by the Upwid scheme. The low ode Peissma scheme is moe sitable fo the lage stide legth, ad the possibility of a itegal step legth has bee impoved i this way, whee =0.6, =0.4, t =19900s, ad D =0.05, which shows a diffeece of 0.94% with D. Whe t =19870s ad D =0.065, D show a diffeece of.77% with D ad pecisio ca meet the eqiemets. Istead, Lac Fiedichs scheme is moe sitable fo small time step. Thogh the step is smalle, the simlatio accacy is ot impoved, sch as whe t =47s ad D =0.097, t is oly 0.1s smalle tha 47.1s, while the elative eo of diffsio coefficiet eaches a satisfactoy level, 5.83%. TABLE 6 RESULTS OF DIFFERENT CALCULATION FORMULATIONS Scheme ame D epessio t epessio Calclatio eslts Dobbis C(1 C) k t t Upwid C(1 C) k t t Implicit C(1 C) k t t Peissma C( 1) k(1 ) C ( 1) t t ( 1) 015. Lac Fiedichs (1 C C ) 5 kk k t t 47.1 Note: 1. I the table k 0.5 ( gh) / C, whee h =4m, Chezy coefficiet C =5, flow ate =0.5m/s, =00m;. Peissma scheme oly coside the low ode coditio, i.e., is ot eqal to 0.5, 1,, ae espectively coespodig to the diffeet vales of,, whee 1 is coespodig to =0.6, ad =0.4, is coespodig to =0.4, ad =

6 Stdy of Civil Egieeig ad Achitecte (SCEA) Volme Isse 1, Mach Coclsios The pape stats fom Upwid scheme, statig the pocess that geeates meical diffsio whe dispesig sspeded load covectio diffsio eqatio with a low level diffeece scheme, ad it ties to simlate the physical diffsio pheomeo sig meical diffsio, simlatio accacy de diffeet paametes, sch as t, ad, ae calclated. Ad by hoizotally compaig diffeet diffeetial schemes we ca get the followig coclsios: 1) Nmeical diffsio of diffeetial scheme ca be sed to simlate the actal physical diffsio pheomeo, bt it has cetai eqiemets o the paamete selectio. Fo sspeded load covectio diffsio eqatio, the simlatio method is to dispese the pe covectio eqatio fist, ad the popose diffsio tem. Sice the coefficiet i fot of diffsio tem is the meical diffsio coefficiet, by assmig which eqals to the physical diffsio coefficiet, we ca get the elatioship eqatio composed of some paametes that meet the simlatio accacy. ) I geeal, i the vaios paametes, oly spatial iteval ad time step t which ae elated to the calclatio ca be chose adomly, ad the selectio of eeds to be compatible with the size of the poject ad the compleity of the boday, so the key is the time step t. By calclatio, this pape has fod the elatioship of t that meets the accate simlatio de vaios diffeetial schemes, bt sally the calclated step sizes ae with decimals, which is ot codcive to calclatio, so a istace is sed to calclate the elative eo betwee the meical diffsio coefficiet ad the physical diffsio coefficiet whe t takes a simila itege vale. It is fod that as the step of lowlevel Peissma scheme is lage, the possibility of a itegal step is geatly iceased, theeby iceasig the calclatio accacy, which is a ecommeded method. 3) Thogh this aticle oly aalyzed sspeded load covectio diffsio eqatio, it ca be speclated that bedload covectio diffsio eqatio, as well as the covectio diffsio pheomea of polltats i the wate, ca be aalyzed sig the same method. Similaly, the method ca be employed to solve the geeal poblems by meas of its iqe speioity of diffeetial eqatios clea mathematical fodatio, simple calclatio ad easy pogammig, which is still favoably compaed to a moe pecise meical method. REFERENCES Gasioowski D. Balace Eos Geeated by Nmeical Diffsio i the Soltio of No liea Ope Chael Flow Eqatios. Joal of Hydology 476 (013): Papadakis Atois P., ad Metaas Adew C. Optimm Mesh Depedet Diffsio Coefficiet Poof fom High to Low Ode Upwid Schemes Utilized i Plasma Dischages. 10th WSEAS Iteatioal Cofeece o EHACʹ11 ad ISPRAʹ11, 3d WSEAS It. Cof. o Naotechology, Naotechologyʹ11, 6th WSEAS It. Cof. o ICOAAʹ11, d WSEAS It.Cof. o IPLAFUNʹ11, 011. Richad Coat, Kt Otto Fiedichs, ad Has Lewy. O the Patial Diffeece Eqatios of Mathematical Physics. Math. A. 100 (198): 3. Xie Jiaheg. Rives Simlatio. Beijig: Wate Powe Pess, Yag Gol. Rives Mathematical Model. Beijig: Ocea Pess, Zheg Bagmi, ad Zhao Xi. Calclatio of Wate Dyamics. Wha: Wha Uivesity Pess,