Journal of Hydrology

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1 Journal of Hydrology 59 (24) Contents lsts avalable at ScenceDrect Journal of Hydrology journal homepage: Hgh resoluton numercal schemes for solvng knematc wave equaton Chunshu Yu, Jennfer G. Duan,2 Department of Cvl Engneerng and Engneerng Mechancs, Unversty of Arzona, USA artcle nfo summary Artcle hstory: Receved 3 July 23 Receved n revsed form 3 June 24 Accepted 2 August 24 Avalable onlne 3 August 24 Ths manuscrpt was handled by Konstantne P. Georgakakos, Edtor-n-Chef, wth the assstance of Ioanns K Tsans, Assocate Edtor Keywords: Knematc wave equaton Godunov-type scheme MacCormack scheme Shock wave Rarefacton wave Ranfall-runoff overland flow Ths paper compares the stablty, accuracy, and computatonal cost of several numercal methods for solvng the knematc wave equaton. The numercal methods nclude the second-order MacCormack fnte dfference scheme, the MacCormack scheme wth a dsspatve nterface, the second-order fnte volume scheme, and the ffth-order fnte volume scheme. These numercal schemes are tested aganst several synthetc cases and an overland flow experment, whch nclude shock wave, rarefacton wave, wave steepenng, unform/non-unform ranfall generated overland flows, and flow over a channel of varyng bed slope. The results show that the MacCormack scheme s not a Total Varaton Dmnshng (TVD) scheme because oscllatory solutons occurred at the presence of shock wave, rarefacton wave, and overland flow over rapdly varyng bed slopes. The MacCormack scheme wth a dsspatve nterface s free of oscllaton but wth consderable dffusons. The Godunov-type schemes are accurate and stable when dealng wth dscontnuous waves. Furthermore the Godunov-type schemes, lke and scheme, are needed for smulatng surface flow from spatally non-unformly dstrbuted ranfalls over rregular terrans usng moderate computng resources on current personal computers. Ó 24 Elsever B.V. All rghts reserved.. Introducton The knematc wave equaton was frst developed by Lghthll and Whtham (955). The equaton s based on the assumptons that the acceleraton term and the pressure gradent term n the momentum equaton are neglgble, so that the energy slope s equal to the bottom slope. The knematc wave model s commonly used to smulate the overland flow (Ponce, 99; Sngh, 2). Henderson (966) showed that natural flood waves behave nearly the same as the knematc wave n steep slopes (S > :2). Vera (93) concluded that the knematc wave equaton can be used on natural slopes wth the knematc wave number, k 5. Ponce (99) compared the knematc wave equaton wth the unt hydrograph as a practcal method of overland flow routng. Sngh (2) concluded that the knematc wave equaton s applcable to surface water routng, vadose zone hydrology, rverne and costal processes, eroson and sedment transport, etc. The knematc wave equaton s a frst-order hyperbolc partal dfferental equaton (PDE). For a hyperbolc equaton, the Correspondng author. E-mal addresses: chunshu@emal.arzona.edu (C. Yu), gduan@emal.arzona.edu (J.G. Duan). Post-doc Research Assocate. 2 Assocate Professor. dsturbance wll travel along the characterstcs of the equaton n a fnte propagaton speed. Ths feature dstngushes the hyperbolc equatons from ellptc and parabolc equatons. On the other hand, the knematc wave equaton also belongs to a knd of equatons called conservaton laws (LeVeque, 22; Toro, 29). Snce the flux term s a nonlnear functon of conservatve varables, the soluton does not propagate unformly but deforms as t evolves. Even the ntal condtons are contnuous and smooth, the hyperbolc conservaton laws can develop dscontnutes n the soluton, for example, shock waves. Both shock and rarefacton waves are the ntrnsc features of hyperbolc equatons. Lghthll and Whtham (955) dscussed the formatons of shock wave and rarefacton wave. Kbler and Woolhser (97) nvestgated the structure and general propertes of shock waves and developed a numercal procedure for shock fttng. Eagleson (97) found that usng non-unform flow depth as ntal condton, non-unform ranfall n the source term, or ncreasng nflows as the boundary condton may cause the formaton of knematc shock wave. Borah et al. (9) presented the propagatng shock-fttng scheme (PSF) to smulate overland flow wth shock waves. Sngh (2) found three factors that affect the shock wave formaton: () ntal and boundary condtons; (2) lateral nflow and outflow, and (3) watershed geometrc characterstcs. Due to the complex geometry, non-unform roughness /Ó 24 Elsever B.V. All rghts reserved.

2 24 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Nomenclature c wave celerty (m/s) f nfltraton rate (mm/h) h flow depth (m) h flow depth at the center of cell (m) h +/2 reconstructed depth at the nterface + /2 of cell (m) h L, h R flow depth at the left and rght of wave front (m) ranfall excess (mm/h) ntensty of ranfall (mm/h) L(h) operator L(h) defned n Eq. (2) ( ) L channel length (m) m exponental n Eq. (9) ( ) n Mannng s roughness coeffcent (sm /3 ) q dscharge per unt wdth (m 2 /s) q L outflow dscharge (m 2 /s) r +/2 dstance from the cell center to the nterface + /2 (m) S shock wave speed (m/s) S bed slope ( ) TV(h) total varaton of flow depth ( ) t smulaton tme (s) t r duraton of ranfall (s) tme of concentraton (s) t c t p v x Subscrpt Superscrpt tme defned n Eq. (3) (s) flow velocty (m/s) downslope dstance (m) spatal ndex n, n þ, and n + and n +, temporal ndces Greek a coeffcent n Eq. (9) ( ) c j lnear weghts ( ) h coeffcent for the dsspatve nterface ( ) e = 6 truncaton error for actual calculatons ( ) Dt tme step (s) Dx space step (m). rh lmted depth gradent of cell ( ) rh L, rh R lmted depth gradent at the left and rght of cell nterface ( ) and non-unform ranfall pattern, t s mpossble to derve a general analytcal soluton for the knematc wave equaton. Sngh (2) summarzed three numercal technques for solvng the knematc wave equaton: () method of characterstc, (2) Lax Wendroff fnte dfference method, and (3) fnte element method. Numercal dffuson and numercal dsperson were observed when usng the fnte dfference schemes (Ponce, 99). Kazezyılmaz- Alhan et al. (25) evaluated several fnte dfference schemes for solvng knematc wave equaton: the lnear explct scheme, the four-pont Pressmann mplct scheme, and the MacCormack scheme. The study (Kazezyılmaz-Alhan et al., 25) found the Mac- Cormack scheme s better than the four-pont mplct fnte dfference scheme for shock capture. However, Kazezyılmaz-Alhan et al. (25) dd not explctly examne the dsperson occurred at the shock and rarefacton waves from non-unformly dstrbuted ranfall. The stablty of the classcal MacCormack scheme at the presence of shock and rarefacton wave remans unknown. Recently, the Godunov-type fnte volume method has been wdely used n solvng shallow water equatons (LeVeque, 22; Toro, 29) because of ts wde applcablty, strong stablty, and hgh accuracy. One of the most popular Godunov-type methods s a second-order, TVD (Total Varaton Dmnshng) scheme, namely the (Monotone Upstream-centered Schemes for Conservaton Laws) scheme (van Leer, 979). The scheme s a hghresoluton scheme because () the spatal accuracy of the scheme s equal to or hgher than second order; (2) the scheme s free from numercal oscllatons or wggles; (3) hgh-resoluton s produced around dscontnutes. In general, the hgh-resoluton schemes are consdered as tradeoffs between computatonal cost and desred accuracy (Harten, 93; Toro, 29). Another popular but relatvely new method s the hgh-order (Weghted Essentally Non-Oscllatory) fnte volume scheme (Shu, 999). Hghorder means the order of accuracy s equal to or hgher than the thrd-order. Accordng to Shu (29), the scheme s sutable for the complcated problems, such as flow havng both shocks and complcated smooth structures (e.g., small perturbaton). Although the computatonal cost of hgh-order scheme can be three to ten tmes than a second-order hgh-resoluton scheme, t s stll preferable because of ts hgh-order accuracy n both tme and space. The applcatons of those two hgh resoluton schemes to solve the knematc wave equaton have not been studed. Whether or not these fnte volume schemes have advantages over the commonly used fnte dfference schemes are examned n ths paper. Ths study compares the Godunov-type fnte volume method usng scheme and scheme wth the fnte dfference method usng MacCormack scheme. The paper s organzed as follows: Secton 2 ntroduces the knematc wave equaton and ts analytcal solutons; Secton 3 dscusses the numercal schemes: the MacCormack scheme, the scheme and the scheme; Secton 4 shows the results of typcal test cases. Fnally, several concludng remarks are gven n Secton Governng equatons The one-dmensonal knematc wave equaton for flows over a slope s gven by (Eagleson, 97; Lghthll and @x ¼ where h s the depth of flow; q s the dscharge per unt wdth; = f s the ran excess; s the ntensty of ranfall; f s the nfltraton rate; t s the tme; x s the downslope dstance. For the overland flow, the dscharge q s defned as: q ¼ ah m where m s the exponental, and a s the coeffcent. For fully turbulent flow, the coeffcents are gven by Ponce (99): a ¼ p ffffffffff 5 S ; m ¼ ð3þ n 3 where n s the Mannng s roughness coeffcent; S s the bottom slope. It s obvous that the flux functon q(h) s a convex functon (Jacovks and Tabak, 996; Toro, 29) because the second order dervatve s postve: d 2 q dh ¼ amðm 2 Þhm 2 > ; for h > ð4þ The analytcal soluton for Eq. () has been found by (Eagleson, 97) n whch the outflow hydrograph s a functon of ranfall ntensty and the tme of concentraton. ðþ ð2þ

3 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Numercal schemes Snce the knematc wave equaton s a nonlnear hyperbolc partal dfferental equaton, dfferent numercal schemes exhbt dfferent amounts of numercal dffuson and dsperson dependng on the nature of schemes. Numercal dffuson often presents tself as the attenuaton of the knematc wave, whle numercal dsperson s responsble for oscllatons or negatve outflows near large surface gradents. To provde comparsons wth classcal fnte dfference schemes, a study of two hgh-resoluton Godunov-type fnte volume schemes, the fnte volume scheme and the fnte volume scheme, s presented n ths paper. All the selected schemes are explct, but dffer n the order of accuracy. The scheme s a second-order scheme whle the scheme s ffth-order. 3.. MacCormack fnte dfference scheme The MacCormack scheme (MacCormack, 23) s a commonly used fnte dfference scheme to solve hyperbolc PDEs. Ths twostep scheme s second-order accurate n both tme and space. Compared to the frst-order scheme, the MacCormack scheme does not ntroduce numercal dffuson n the soluton. However, the numercal dsperson can be ntroduced n the regon of large surface gradents. The MacCormack scheme s also a varaton of the two-step Lax Wendroff scheme. It ncludes two steps: a predctor step followed by a corrector step. The predctor step uses forward dfference approxmatons whle the corrector step uses backward dfference approxmatons for spatal dervatves. The order of dfferencng can be reversed from tme step to tme step (.e., forward/ backward dfferencng followed by backward/forward dfferencng). The tme step used n the predctor step s Dt n contrast to Dt/2 used n the corrector step. The dscretzed knematc wave equaton usng the MacCormack scheme s below: h nþ Predctor step: ¼ h n Dt Dx ðqn þ qn Þþ Dt Corrector step: h nþ ¼ 2 hn þ h nþ 2 Dt Dx qnþ q nþ þ Dt ð5aþ ð5bþ where subscrpt s the spatal ndex; superscrpt n, n þ, and n + are the temporal ndces; Dt s tme step; Dx s the space step. The classcal MacCormack scheme creates spurous oscllatons at the fronts of shock and rarefacton waves (Garca-Navarro et al., 992; Macchone and Morell, 23). A remedy s to smooth out the oscllatons usng the dsspatve nterface (DI) after the corrector step (Abbott and Mnns, 99). Mathematcal formulaton for the DI s as follows: >< >: h nþ; h nþ; h nþ; ¼ hh nþ þ ¼ hh nþ þ ¼ð hþh nþ þð 2hÞhnþ þ hh nþ ; for nternal nodes þð hþhnþ ; for left boundary node þ hh nþ ; for rght boundary node where h nþ; s the averaged soluton; h s the coeffcent for the dsspatve nterface fnte volume scheme The scheme was ntroduced by van Leer (979). It s the frst second-order TVD Godunov-type fnte volume scheme. MUS- CL uses pecewse lnear approxmaton to reconstruct the depths at the nterfaces of cells: h þ=2 ¼ h þ rh r þ=2 ð6þ ð7þ where h s flow depth at the center of cell ; h +/2 s the reconstructed depth at the nterface + /2 of cell; rh s the lmted depth gradent of cell ; r +/2 s the dstance from the cell center to the nterface + /2. The lmted gradents can be calculated by several slope lmters (e.g., Mnmod, Superbee, etc.). The van Leer lmter (Causon et al., 2; van Leer, 974) s used n the study: rh ¼ rh Ljrh R jþjrh L jrh R jrh L jþjrh R j where rh L ¼ h h Dx ðþ and rh R ¼ h þ h. The local Lax Fredrchs (LLF) Dx method (LeVeque, 22) s used to calculate the fluxes across the cell nterfaces: Q =2 ¼ 2 ½q þ q a =2 ðh h ÞŠ a =2 ¼ maxðjc j; jc jþ ð9þ ðþ where c s the celerty and defned n Eq. (9). Based on the calculated numercal fluxes across the nterfaces, the soluton can be updated by a tme marchng method. Instead of usng the predctor corrector method of -Hancock scheme, the second-order TVD Runge Kutta method (Gottleb and Shu, 99) s employed n the study to dscretze the tme dervatve. The second-order TVD Runge Kutta method (RK2) can be wrtten as: h ðþ ¼ h n þ Dt L ðh n Þ h nþ ¼ 2 hn þ 2 hðþ þ Dt 2 LðhðÞ Þ where the operator L(h) s defned as: ðþ LðhÞ ¼ Dx X QðhÞþ ð2þ For a TVD Runge Kutta method, t s guaranteed that each ntermedate soluton also satsfes the TVD crtera to avod spurous oscllatons n the soluton. Because of ths, t s a better choce to ncorporate a TVD Runge Kutta method for solvng hyperbolc equaton fnte volume scheme The frst scheme was provded by Lu et al. (994). Jang and Shu (996) presented a general framework to construct arbtrary hgh order schemes. Detals about schemes can be found n Shu (999, 29). Instead of usng the pecewse lnear reconstructon procedure n the scheme, the fnte volume scheme uses the reconstructon procedure to obtan an approxmated functon value at the cell nterface. The reconstructon procedure conssts of four steps: Step. Calculate the smoothness ndcators: >< >: b ¼ 3 2 ðh 2 2h þ h Þ 2 þ 4 ðh 2 4h þ 3h Þ 2 b 2 ¼ 3 2 ðh 2h þ h þ Þ 2 þ 4 ðh h þ Þ 2 b 3 ¼ 3 2 ðh 2h þ þ h þ2 Þ 2 þ 4 ð3h 4h þ þ h þ2 Þ 2 ð3þ Step 2. Calculate the thrd-order approxmatons at cell nterfaces: h ðþ þ=2 ¼ h h 6 þ h 6 >< h ð2þ þ=2 ¼ h 6 þ 5 h 6 þ h 3 þ ð4þ >: h ð3þ þ=2 ¼ h 3 þ 5 h 6 þ h 6 þ2 Step 3. Calculate the nonlnear weghts: ~w j w j ¼ ; wth ~w j ¼ ; j ¼ ; 2; 3 ð5þ ~w þ ~w 2 þ ~w 2 3 ðe þ b j Þ c j

4 26 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) where e ¼ 6 for actual calculatons; c j s the lnear weghts, and s gven by: c ¼ 6 ; c 2 ¼ 5 ; c 3 ¼ 5 6 ð6þ Step 4. Calculate the ffth-order approxmaton as a convex combnaton of the three thrd-order approxmatons: h þ=2 ¼ w h ðþ þ=2 þ w h ð2þ þ=2 þ w h ð3þ þ=2 ð7þ The dervatve n tme s dscretzed by the thrd-order TVD Runge Kutta method (Gottleb and Shu, 99), whch s a threestep method: h ðþ ¼ h n þ Dt Lðh n Þ h ð2þ ¼ 3 4 hn þ 4 hðþ þ Dt 4 LðhðÞ Þ h nþ ¼ 3 hn þ 2 3 hð2þ þ 2 Dt 3 Lðhð2Þ Þ where the operator L(h) s gven by Eq. (2). 4. Results ðþ Ths secton evaluates the selected schemes n a varety of flow condtons because a gven scheme does well for a test case does not guarantee t wll do well for another test case. In order to evaluate the performances of these schemes, t s necessary to choose test cases wth exact solutons. For ths reason, the one-dmensonal propagatons of a shock wave and a rarefacton wave are selected. Then the behavors of these schemes are tested by a wave steepenng example. In addton, two synthetc ranfall-runoff cases and one overland flow over a vared slope are selected to demonstrate the applcablty of those schemes. Fnally, a real expermental case s smulated to test the performance of these schemes. The followng abbrevatons are used n the fgures: MUS- CL stands for the results by usng the scheme; for the scheme; for the MacCormack scheme; for the MacCormack scheme wth a dsspatve nterface. h =.25 s the coeffcent for the dsspatve nterface used n all the test cases. 4.. Shock wave The shock wave and rarefacton wave are the ntrnsc features of the hyperbolc equatons, and thus the knematc wave equaton. The celerty of knematc wave s defned as (Lghthll and Whtham, 955): c ¼ amh m ¼ mv ð9þ where v s flow velocty. Consder the followng ntal-value problem for the knematc wave equaton: hðx; Þ ¼ h L; f x < x ð2þ h R ; f x > x If we assume h L > h R, we wll have c L > c R. Ths means that a shock wave wll arse. The exact shock wave soluton for the knematc wave equaton s: hðx; tþ ¼ h L; forðx x Þ=t < S ð2þ h R ; forðx x Þ=t > S where S s the shock wave speed. For the knematc wave equaton, accordng to the Rankne-Hugonot jump condton (Toro, 29), S s equal to: S ¼ a hm R hm L h R h L ð22þ In the study, the followng parameters are used: the channel length L =. m; the bed slope S =.6; the Mannng s coeffcent n =.25 s/m /3 ; the smulaton tme t = 3.5 s; the tme step Dt =. s; the grd spacng Dx =. m. The ntal condton s: h L =. m and h R =.5 m. The boundary condtons are zero-depth gradent at both the entrance and the ext of the channel. The flow depth profles calculated by the numercal schemes and the exact soluton are plotted n Fg.. The results show that the numercal dffuson for flow depth near the front of shock wave s small for the and schemes. But the results of flow depth from the MacCormack scheme are oscllatory. The wggled surface oscllatons are largest at the dscontnuous wave front, and the solutons eventually dverge. No oscllatons are present usng the MacCormack scheme wth the dsspatve nterface. But, the results are not accurate because the dsspatve nterface has smoothed out the sharp front of the shock wave. To determne f a numercal scheme wll ntroduce oscllatons n the soluton, the total varaton defned n Eq. (23) can be used. TVðhÞ ¼ X ¼ jh h j ð23þ where TV(h) s the total varaton of flow depth, s the ndex of computatonal cells. If the total varaton s ncreasng wth tme, the numercal scheme wll ntroduce oscllatons; otherwse, t preserves the monotoncty of ntal monotone functons (LeVeque, 22). To avod oscllatory solutons, the total varaton needs to decrease or reman constant wth tme. The total varatons of flow depth from those schemes are shown n Fg. 2, whch ndcate that the total varaton ncreases wth tme usng the MacCormack scheme, whle t remans constant for, and schemes. Ths mples that the MacCormack scheme wll ntroduce oscllatons at the presence of shock waves, but, and schemes can preserve the functonal propertes of the shock wave. Therefore, the solutons usng the MacCormack scheme are.2..6 EXACT Channel Dstance (m) EXACT Channel Dstance (m) Fg.. Results of shock wave test case: global vew; local vew.

5 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Total Varaton (m) EXACT.5.4 oscllatory, and those usng the scheme are naccurate for capturng the shock wave Rarefacton wave Reconsder the ntal-value problem descrbed n Eq. (2), and assume h L < h R, we have c L < c R. Ths tme, nstead of generatng a shock wave, a rarefacton wave s generated near the dscontnuty snce the celerty at the head of the dscontnuty s greater than that at the tal and, consequently, the dscontnuty contnually expands as t propagates. For the knematc wave equaton, the exact soluton of rarefacton wave s: >< h ¼ >: Tme (s) Fg. 2. Calculated total varatons for the shock wave test case. h L ; for x x t < c L h L þð x x c t L Þ h R h L c R c L ; for c L 6 x x 6 c t R h R ; for x x t > c R ð24þ The parameters used n the rarefacton test case are summarzed as: the channel length L =. m; the bed slope S =.6; the Mannng s coeffcent n =.25 s/m /3 ; the smulaton tme t = 3. s; the tme step Dt =. s; and the grd spacng Dx =. m. The ntal condton s: h L =.5 m and h R =. m. The boundary condtons are zero-depth gradent at both ends of the channel. The calculated flow depth profles are plotted n Fg. 3. The results are smlar to the shock wave case: the MacCormack scheme generated notable oscllatons at the tal of the rarefacton wave. Taylor et al. (972) tested the MacCormack scheme for solvng the Burgers equaton and found that the MacCormack scheme s unstable for rarefacton wave under some condtons. Ths study fnds that the same phenomenon occurred to the knematc wave equaton. Snce the oscllatons occur at the tal of the rarefacton wave, where flow depth s smaller, unrealstc negatve flow depths can be nduced at the tals of rarefacton waves. Ths can also be proved by calculatng the total varaton of flow depth as shown n Fg. 4. The total varatons from, and schemes are constant. But the values from the MacCormack scheme ncrease suddenly to a peak value and then gradually decrease to a constant value (about.9) greater than the ntal total varaton (=.5). Although the total varaton shows a decreasng trend, the scheme cannot keep the total varaton not to ncrease wth tme for the entre smulaton tme. Therefore, oscllatons occur at the dscontnuous front, the tal of rarefacton wave. Therefore, for the rarefacton wave, the MacCormack scheme s not as stable as the, or scheme. Both the shock and rarefacton wave test cases suggest that the MacCormack scheme s not a Total Varaton Dmnshng (TVD) scheme, and the oscllatons wll be generated near the dscontnuous wave Channel Dstance (m) EXACT fronts. Although the scheme has elmnated spurous oscllatons n the soluton, t also smears out the dscontnuous wave fronts by ntroducng redundant numercal dffuson to the soluton. Therefore, the scheme and schemes should be used to smulate unsteady flows wth shock and rarefacton waves Wave steepenng Channel Dstance (m) Fg. 3. Results of rarefacton wave test case: global vew; local vew. Total Varaton (m) Tme (s) Fg. 4. Calculated total varaton for the rarefacton wave test case. One of the promnent features of the hyperbolc equaton s that dscontnutes wll be generated even the ntal water surface s smooth. So t s mportant for a scheme to preserve the stablty

6 2 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) and sharpness of dscontnuous fronts n a smulaton. Ths test case s to test ths behavor of these schemes, especally the ablty of ant-dffuson. Here are the parameters n the test case: the channel length L =. m; the bed slope S =.6; the Mannng s coeffcent n =.25 s/m /3 ; the smulaton tme t = 3.5 s; the tme step Dt =. s; and the grd spacng Dx =. m. The ntal condton s: h h ¼ max :5; : ðx :Þ 2 ð25þ Ths ntal condton wll create a parabolc perturbaton n the channel as shown n Fg. 5. The boundary condtons are perodc flow depths at both ends of the channel. The parabolc perturbaton wll produce a wave steepenng effect n the doman (Henderson, 966; Toro, 29). The reason s that the celerty of the perturbaton s faster than the ambent flud. The head of the perturbaton s a compressve regon whle the tal of the perturbaton s an expansve regon. Ths perturbaton s a combnaton of a shock wave (head) and a rarefacton wave (tal). Ths scenaro s detrmental to the MacCormack scheme because the oscllatons occurred behnd the shock wave wll drown out the parabolc perturbaton mmedately (Fg. 6). Consequently, the MacCormack scheme cannot converge to a stable soluton, but amplfed oscllatons. Ths further proves that the MacCormack scheme s an unstable scheme for flows of dscontnuous waves. The results of, and schemes are plotted n Fg. 7. After 3.5 s, the fronts of the perturbaton reman sharp usng the and schemes, whereas the results from the scheme cannot preserve the sharp front because of the numercal dffuson Unform ranfall-runoff overland flow The unform ranfall-runoff overland flow can be solved by the method of characterstcs (Eagleson, 97; Kazezyılmaz-Alhan et al., 25). At a gven ranfall excess n a specfed duraton, the outflow hydrograph (q = q L, x = L) can be solved analytcally. Assumng a constant ranfall excess ( ) and an ntal zero flow depth, h ¼ ðt ¼ and 6 x 6 LÞ and the boundary condton at the channel entrance, h ¼ ðt > and x ¼ Þ ð26þ ð27þ The solutons of two possble outflow hydrographs are summarzed below: Case (t r P t c ): Channel Dstance (m) Fg. 6. Unstable result of the wave steepenng test case Channel Dstance (m) Fg. 7. Stable results of the wave steepenng test case. >< h L ¼ t; for t 6 t c q L ¼ ah m L ;to solve h L use : h L ¼ t c ; for t c < t 6 t r >: L ¼ ah m L ½h L þ mðt t rþš; for t > t r ð2þ Case 2 (t r < t c ): >< h L ¼ t; for t 6 t r q L ¼ ah m L ;to solve h L use : h L ¼ t r ; for t r < t 6 t p >: L ¼ ah m L ½h L þ mðt t rþš; for t > t p ð29þ where t r s the duraton of ranfall, and t c s the tme of concentraton: 2 t c ¼ L m a! =m ð3þ Channel Dstance (m) Fg. 5. Intal condton of wave steepenng case. and t p s defned as: t p ¼ t r þ t c t r m ; t c ¼ L ah m Lr ; h Lr ¼ t r ð3þ In Kazezyılmaz-Alhan et al. (25) s hypothetcal experment, the duraton of ranfall s longer than the tme of concentraton (t r P t c ). The expermental parameters are summarzed here: the channel length L = 2. m; the bed slope S =.6; the Mannng s coeffcent n =.25 s/m /3 ; the duraton of ranfall t r =.5 h; the ranfall excess = 5. mm/h; smulaton tme t = h; the tme step Dt =. s; and the grd spacng Dx =.3 m. The ntal condton s h =. m. The boundary condtons are zero-depth gradent at the outlet and zero depth at the nlet.

7 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Fg. plots the outflow hydrographs calculated by the numercal schemes and the exact soluton calculated by Eq. (2). Snce there s no dscontnuty/perturbaton n the doman, the results obtaned by the MacCormack, and schemes are very close to the exact soluton. The results usng the scheme showed remarkable numercal dffusons than those usng other schemes. The comparsons of the peak of the hydrographs show that the scheme yelds the closest results to the exact soluton wthout any oscllaton. When the outflow dscharge reaches the peak, flow n the channel reaches steady state. To check f the mass s conserved n the smulaton, the dfferences between the total outflow volume and the total ranfall volume are calculated, and then non-dmensonalzed by the total ranfall volume. The percentage flow dfferences relatve to the total ranfall are.65% for the MacCormack scheme,.59% for the scheme,.35% for the scheme, and.3% for the scheme, respectvely. All schemes preserve mass conservaton very well at the absence of dscontnuous waves. The scheme and scheme are slghtly more conservatve than the MacCormack scheme Steady non-unform ranfall-runoff overland flow To compare the stablty of those schemes, a steady non-unform ranfall-runoff case, Case I n Moramarco and Sngh (22), was smulated. The smulated plane s m long wth a slope of.5 3, and the Mannng s roughness coeffcent s.3 s/m /3. The followng steady non-unform ranfall excess s used n ths case (Fg. 9): ð 5 4:43; x 6 5 m m=sþ ¼ ð32þ :; x > 5 m Ranfall Intensty (m/s) 6 x Channel Dstance (m) Fg. 9. Non-unform ranfall dstrbuton. The smulated results are showed n Fgs. and. The depth of runoff generated by the non-unform ranfall s not smooth and a shock wave s generated near the front of runoff. Oscllatons occurred n the soluton of the MacCormack scheme. Solutons by the, and schemes are stable, and free of oscllatons. But, only the results usng the and schemes can capture the sharp front of shock wave generated by the runoff. The computaton s run wth a DELL XPS notebook (Intel 5 M GHz CPU wth 4 GB memory). The programs are developed usng Matlab 2b runnng on Mcrosoft Wndows 7. OS. For ths smulaton, the CPU tmes usng the MacCormack,,, and schemes are 3.4 s, 3.7 s, 4. s, and 27.3 s, Dscharge (m 3 /s) Dscharge (m 3 /s) EXACT Tme (s) EXACT x Channel Dstance (m) x Tme (s) Channel Dstance (m) Fg.. Hydrograph of the unform ranfall-runoff case: global vew; local vew. Fg.. Results of non-unform ranfall-runoff test case: global vew; local vew.

8 3 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Dscharge (m 3 /s) Flow Dscharge (m 2 /mn) Dscharge (m 3 /s) Tme (s) Tme (s) Flow Dscharge (m 2 /mn) Tme (s) Tme (s) Fg.. Hydrographs of non-unform ranfall-runoff test case: global vew; local vew. Fg. 2. Hydrographs of the non-unform overland slope test case: global vew; local vew. respectvely. As expected, the MacCormack scheme runs the fastest, whle the scheme s two tmes slower than the scheme Ranfall-runoff over non-unform overland slope Snce overland flow always travels n channels of varyng slopes, ths requres the numercal scheme for solvng the knematc wave equaton to be vald for changng bed slopes. Pror test cases have shown that the MacCormack scheme s unstable for flows wth dscontnuous waves (e.g., shock, rarefacton). Therefore, the MacCormack scheme s not applcable to non-unform ranfall on a unform slope. To verfy ts applcablty on a changng slope, ths study hypotheszes an experment of ranfall and overland flow n a watershed havng the same expermental condtons as n Kazezyılmaz-Alhan et al. (25) except the upstream half of the channel has a slope dfferent from the downstream half (S =.6). Ths study adopted S =.6 as the bed slope for the upstream half of the channel. The smulated water surfaces are shown n Fg. 2. As the slope n the upstream half channel ncreases to.6, ten tmes of the downstream slope, the MacCormack scheme generated severe oscllatons. Ths test verfes that the MacCormack scheme s not only unstable for shock and rarefacton waves but also for flows on rapdly varyng slopes Overland flow experment (Schreber, 97) Ths laboratory experment was conducted n the Eroson Laboratory, Palouse Conservaton Feld Staton, USDA-ARS-SWC (Schreber, 97). The experment was conducted on a sloped plane to mmc one dmensonal overland flow. Durng the experment, the ranfall s generated by an array of nozzles. The runoff from the plane was draned nto a tank and the dscharge was measured by a stran gauge. The length of the plane s 4. m. The plane slope s.465, and the Mannng s roughness coeffcent s.25 s/m /3. The ntensty of ranfall s 27 mm/h wth duraton of 4. mn. For the model, the smulaton tme s. mn. The upstream boundary condton s zero flux, the same as the sold wall boundary condton. The downstream boundary condton s defned by the characterstc equaton (Eq. 9). The smulated hydrographs by usng the above mentoned schemes are compared wth the measured ones n Fg. 3. Fg. 3 shows the smulated results are nearly the same usng dfferent schemes. A detaled plot n Fg. 3 shows the MacCormack scheme over-predcted the dscharge. However, none of the smulated results matched the expermental observatons. Ths attrbutes to the assumpton of knematc wave equaton, whch has a smaller celerty than the dynamc wave, and unable to capture the shock wave front. Therefore, the proposed and solver for the knematc wave equaton s vald only for overland flow when the knematc wave assumpton s vald. To evaluate the accuracy of those numercal schemes, the rootmean-square error (RMSE) of three test cases,.e. the shock wave case, the rarefacton wave case, and the ranfall-runoff case, are calculated by comparng to the exact solutons n Table. The RMSE showed that the MacCormack scheme wth the dsspatve nterface has the maxmum error, although ts results are free of oscllaton. Ths mples that the dsspatve nterface brought n consderable numercal dffusons to the solutons. On the other hand, the and the scheme are not only free of oscllaton, but also yelded more accurate results than the MacCormack scheme. In summary, the testng cases lsted n Sectons demonstrated the lmtatons of the MacCormack scheme for solvng the

9 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Dscharge (m 3 /s) x Measured scheme, the oscllatons serously affect the stablty of the soluton, and make the soluton dverge. The dsspatve nterface can elmnate spurous oscllatons seen n the MacCormack scheme, but t also brngs n redundant numercal dffusons to soluton. On the other hand, the and schemes consstently perform better than the MacCormack scheme. There s no oscllaton n the solutons usng the scheme or the scheme. The hgh-order scheme shows the best resoluton power n all test cases. Although the computatonal costs are hgher than that of the MacCormack scheme, to ensure numercal stablty, the and schemes are the stable and accurate technques for solvng the knematc wave equaton. Dscharge (m 3 /s) Tme (mn) x -5 knematc wave equaton and proved the valdty of hgh resoluton schemes, the and schemes. Although the MacCormack scheme has been wdely used to solve the knematc wave equaton, t should be avoded at the presence of shock/rarefacton waves, or flow over rapdly varyng sloped land surfaces. The dsspatve nterface for the MacCormack scheme resulted n excessve numercal dffusons, and s not accurate for solvng unsteady flows wth dscontnuous waves. Instead, the and the scheme should be used n spte of the slght ncrease of computaton cost. 5. Conclusons Tme (mn) Fg. 3. Hydrographs of the Schreber (97) overland flow experment: global vew; local vew. Table Calculated root-mean-square errors (RMSE). Test case Shock wave Rarefacton wave Ranfall-runoff In ths study, the applcablty of four numercal schemes (Mac- Cormack scheme, MacCormack scheme wth the dsspatve nterface, scheme and scheme) to solve the knematc wave equaton s nvestgated. The schemes are examned by usng several test cases: shock wave, rarefacton wave, wave steepenng, overland flow from unform or non-unform ranfall, overland flow over varyng bed slopes. The results show that oscllatons appeared n the solutons usng the MacCormack scheme when the shock and rarefacton waves are present, or bed slopes changng rapdly. Snce the MacCormack scheme s not a TVD Acknowledgements The authors are grateful for research fundng provded by Unted States Natonal Scence Foundaton Award EAR to the Unversty of Arzona. The authors wsh to acknowledge n partcular Chunyan Gao for valuable dscussons on the knematc wave equaton. The authors also would lke to thank the revewers and the edtors for ther constructve comments. References Abbott, M.B., Mnns, A.W., 99. Computatonal Hydraulcs. Ashgate, Aldershot, Brookfeld, USA, vol. xv, p Causon, D.M., Ingram, D.M., Mngham, C.G., Yang, G., Pearson, R.V., 2. Calculaton of shallow water flows usng a Cartesan cut cell approach. Adv. 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10 32 C. Yu, J.G. Duan / Journal of Hydrology 59 (24) Toro, E.F., 29. Remann Solvers and Numercal Methods for Flud Dynamcs: A Practcal Introducton. New York, Sprnger, Dordrecht, vol. xxv, p van Leer, B., 974. Towards the ultmate conservatve dfference scheme. II. Monotoncty and conservaton combned n a second-order scheme. J. Comput. Phys. 4 (4), van Leer, B., 979. Towards the ultmate conservatve dfference scheme. 5. 2nd-Order sequel to Godunovs method. J. Comput. Phys. 32 (), 36. Vera, J.H.D., 93. Condtons governng the use of approxmatons for the Sant-Venant equatons for shallow surface-water flow. J. Hydrol. 6 ( 4), 43 5.