NUMERICAL MODELLING OF TSUNAMI WAVE EQUATIONS MASTER OF SCIENCE IN MATHEMATICS

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1 NUMERICAL MODELLING OF TSUNAMI WAVE EQUATIONS A thess submtted n partal fulfllment of the requrements for the award of the degree of MASTER OF SCIENCE IN MATHEMATICS submtted by HARI SHANKAR SHAW Roll No-412MA2085 under the supervson of Prof. SNEHASHISH CHAKRAVERTY DEPARTMENT OF MATHEMATICS NIT ROURKELA ROURKELA MAY 2014

2 NATIONAL INSTITUTE OF TECHNOLOGY ROURKELA DECLARATION I hereby declare that the work whch s beng presented n the report enttled NU- MERICAL MODELLING OF TSUNAMI WAVE EQUATIONS n partal fulfllment of the requrement for the award of the degree of Master of Scence, submtted n the Department of Mathematcs, Natonal Insttute of Technology, Rourkela s an authentc record of my own work carred out under the supervson of Prof. S. Chakraverty. The matter emboded n ths has not been submtted by me for the award of any other degree. May, 2014 (Har Shankar Shaw)

3 CERTIFICATE Ths s to certfy that the proect report enttled NUMERICAL MODELLING OF TSUNAMI WAVE EQUATIONS submtted by Har Shankar Shaw to the Natonal Insttute of Technology Rourkela, Odsha for the partal fulfllment of requrements for the degree of master of scence n Mathematcs s a bonafde record of revew work carred out by hm under my supervson and gudance. The contents of ths report, n full or n parts, have not been submtted to any other nsttute or unversty for the award of any degree or dploma. May, 2014 (Prof. S. CHAKRAVERTY) Head of Department Department of Mathematcs NIT Rourkela (Har Shankar Shaw)

4 ACKNOWLEDGEMENTS I am extremely ndebted to be nvolved n a challengng research proect lke Numercal Modellng Of Tsunam Wave Equatons. Ths proect ncreased my thnkng and understandng capablty as I started the proect from scratch. I would lke to express my greatest grattude and respect to my supervsor Prof. Snehashsh Chakraverty, for hs valuable suggestons, excellent gudance and endless support. He s not only a wonderful supervsor but also a genune person. I consder myself extremely lucky to be able to work under gudance. Actually he s one of such genune person for whom my words are not enough to express. I am very much grateful to Prof. Sunl Kumar Sarang, Drector, Natonal Insttute of Technology, Rourkela for provdng excellent facltes n the nsttute for carryng out research. I would lke to express my sncere thanks to PhD scholars especally Sukant Nayak and Karan Kr. Pradhan for ther help and precous suggestons to perform the proect work. I am very much thankful to them for gvng hs valuable tme for me. I would lke to express my thanks to all the faculty members, all my classmates, all staffs of Mathematcs department for makng my stay n N.I.T. Rourkela a pleasant and memorable experence and also gvng me absolute workng envronment where I unleashed my potental. I owe a grattude to God and my parents for ther blessngs and nspraton.

5 ABSTRACT Ths report nvestgates the modellng of tsunam wave usng one dmensonal shallow water equatons (SWEs) by numercal methods namely fnte dfference method (FDM) and fnte volume method (FVM). We have used one dmensonal SWEs to model the water wave propagaton.e. we study the varaton of water surface elevaton wth fnte dstance. We obtaned the SWEs from Euler s equaton of mass and momentum assumng a long wave approxmaton. Frst of all we approxmate the SWEs usng FDM and then by FVM for showng the behavour of water surface elevaton wth dstance. After approxmatng the SWEs usng both the numercal method, results have been shown usng dfferent schemes vz. FDM as well as FVM. Moreover, n actual practce, we may have ncomplete nformaton about the varables beng a result of errors n modellng, observatons, or by applyng dfferent ntal as well as boundary condtons etc. Rather than the partcular value of water surface we may have only the bounds of the values. These bounds may be gven n term of nterval. Thus we have developed nterval fnte volume method (IFVM) also for approxmatng one dmensonal SWEs to model tsunam wave wth uncertan (nterval) parameter. Next, numercal results have been shown usng IFVM. Then a comparson study has been nvestgated to compare the results of both the method.e FDM and FVM. Fnally all computed results are shown n terms of tables and plots. v

6 Contents 1 Intoducton 1 2 One dmensonal shallow water equatons (SWEs) 4 3 Fnte dfference method (FDM) for solvng one dmensonal SWEs Intoducton to fnte dfference method (FDM) Dfferent schemes of FDM Explct scheme Sem-mplct scheme Implct scheme Tsunam wave approxmaton Fnte volume method (FVM) for solvng one dmensonal SWEs Introducton to fnte volume method (FVM) Dfferent schemes of FVM Upwnd nterpolaton Central dfference (CD) nterpolaton Tsunam wave approxmaton Interval fnte volume method (IFVM) Numercal results Numercal results usng FDM Numercal results usng FVM and IFVM Comparson between FDM and FVM, concluson and future drectons 23 References 26 v

7 1 Intoducton Tsunams are generated by the movement of sea bottom due to long waves of earthquakes. The mpulsve sea floor movement n the earthquakes regon causes the water surface regon nstantaneously as dscussed n [1. The sudden gan n potental energy converts to knetc energy by the gravtatonal force whch serves as the restorng force of the system. Generally, tsunams are treated as shallow water waves. Tsunam s a Japanese word that s the combnaton of two words : tsu means harbor and nam means wave. Therefore, tsunam lterally means harbor waves. The word was orgnally created to descrbe large ampltude oscllatons n a harbor under the resonance condton gven n [2. The most common cause of tsunam are under sea earthquakes. In ths report we have used shallow water equatons ( SWEs ) to model water wave propagaton n one dmenson. We obtan the SWEs from Euler s equaton of mass and momentum consderng a long wave approxmatons gven n [1, [3. SWEs state the propagaton of water waves whose wave length s much longer than the depth of water. Therefore, we have modeled tsunam wave usng SWEs. Shallow Water Equatons (SWEs) Shallow Water Equatons ( SWEs ) are a system of hyperbolc partal dfferental equatons ( PDEs ) governng the flow of flud n the rvers, channels, oceans and costal regons. We have nvestgated SWEs from mass and energy conservaton prncple expressed n the Naver-Stokes equatons. SWEs gve the dea about the flow of water waves, especally those water wave whose wave length s much longer than the depth ( basn ) of water. The wavelength of tsunam waves are far longer than the normal waves. A tsunam wave ntally resembles a rapdly rsng tdes for ths reason they are often referred to as tdal waves. The average depth [3 of ocean nearly 5 Km, whch s compared wth the wavelength long waves or tsunams, whch may exceed 100 Km. Although the mpact of tsunams s lmted to coastal areas, ther destructve power can be enormous and they can affect entre ocean basns; n 2004 Indan Ocean tsunam was among the deadlest natural dsasters n human hstory wth over 230,000 people klled n 14 countres borderng the Indan ocean. One can get SWEs by neglectng the bottom frcton and assumng long wave approxmatons from the Euler equatons of mass and momentum. SWEs are used to model Dam Beaks, strom surges, solute transport, rver flows, economcal model etc gven n [3. The problem wth SWEs s, they are dffcult to model n dry areas where water s not present. SWEs are only defned n wet regons. Thus for these type of equatons we need actually to deal wth movng boundary problems. How do SWEs arse? SWEs are nvestgated by Naver-Stokes (N-S) equatons, whch descrbe the moton of flud. Also, the N-S equatons are derved from the equatons of conservaton 1

8 of mass and lnear momentum dscussed by Imamura [1. So, from the translaton moton of flud element and neglectng the vertcal acceleraton, the equatons of mass conservaton and momentum n one dmensonal problem are descrbed as follows [1 u + w x z = 0 u + u u + w u P + (1/ρ) = 0 t x z x w + u w + w w P + g + (1/ρ) = 0 t x z z where, x s the horzontal axs and z s the vertcal axs, t s tme, η s the water surface elevaton, h s basn depth, u, w are the veloctes of flud n the x and z drectons respectvely, and g s the acceleraton due to gravty. Fnally after takng the dynamcs and knetc condtons at surface and bottom [3, we get the Shallow Water Equatons (SWEs). Mathematcal modelng plays a vtal role n the area of tsunam scence, such as n the area of scentfc studes for tsunam propagaton and ntaton. Ths thess nvestgates the numercal soluton of one dmensonal SWEs usng numercal methods namely Fnte Dfference Method (FDM) and Fnte Volume Method (FVM). FVM s one of the most useful method for modelng the SWEs, long waves, radatve transfer, etc. FVM s wdely used n engneerng, flud mechancs, petroleum engneerng, computatonal flud dynamcs, heat and mass transfer, etc. The most mportant feature of ths method s that numercal flux s conserved from one dscretzaton cell to ts neghbor. Ths feature makes FVM qute attractve for modelng problems n flud mechancs, heat transfer and sem conductor devce smulaton. FVM s a method of representng and evaluatng the partal dfferental equatons to algebrac equatons. In ths method we calculate the values at dscrete places on meshed geometry as n fnte dfference method (FDM) or fnte element method (FEM). Fnte (control) refers to small volume surroundng each node pont on a mesh. Then we used nterval fnte volume method (IFVM) to the SWEs and have nvestgated the varaton of Tsunam wave. The shallow water equatons ntroduced n [6 s very commonly used for numercal soluton. Modellng of tsunam wave has been solved usng lnear Leap-frog method by Imamura, Yalcner [1. A detaled study of SWEs s gven n IUGG/IOC tme proect [3 for numercal methods n tsunam smulaton. A numercal modellng on one dmensonal and two dmensonal SWEs usng FDM dscussed n Junbo Park [4 and added the numercal smulaton of wave propagaton. A lot of research work on one and two dmensonal convecton-dffuson problems, [8 has been solved usng fnte volume scheme by Versteeg and Malalasekera. Cebec, et al. [9 proposed real lfe problems usng FVM. In ths report our man am s to develop an effcent numercal method for solvng SWEs to model Tsunam wave. Generally, the values of varables or propertes are (1) 2

9 taken as crsp but n actual case the accurate (crsp) values may not be obtaned. To overcome the vagueness we use nterval n place of crsp values. So, next am s to study the nterval fnte volume method (IFVM). The IFVM has been developed here to study the varaton of tsunam waves. In ths report we frst dscuss the ntroducton of tsunam wave and shallow water waves and ther orgns n secton 1 and 2. A detaled study of one dmensonal SWEs usng dfferent schemes of FDM has been done n secton 3. Then we have solved one dmensonal SWEs usng FVM wth dfferent schemes of nterpolaton and varaton for FVM and IFVM n secton 4. Secton 5 deals wth the numercal results for one dmensonal SWEs usng FDM and FVM. A comparatve numercal results usng FDM and FVM has been nvestgated n secton 6. Also concluson and future work has been ncluded n secton 6 and fnally references are cted. 3

10 2 One dmensonal shallow water equatons (SWEs) The shallow water Eqs. n one dmenson s gven by [1 η t + M x = 0 (mass conservaton law) where, M t + gd η t = 0 (momentum conservaton law) (2) η = Water surface elevaton M = Dscharge flux n the postve x-drecton g = Acceleraton due to gravty D = Total thckness of water h = Basn depth of water Thus, D = η + h Fg. 1 shows the behavour of one dmensonal shallow water equatons, where vertcal axs represents water surface elevaton and horzontal axs represents dstance. Fgure 1: 1-D shallow water model The Eqs. n (2) are coupled frst order partal dfferental equatons whch can be uncoupled to produce two second order partal dfferental equatons [4 whch are gven as follows 2 M t 2 2 η t 2 = g(η + h) 2 M x 1 M M 2 (η + h) x t = g(η + h) 2 η x 2 + g ( ) 2 η + g h x x η x (3) 4

11 3 Fnte dfference method (FDM) for solvng one dmensonal SWEs In ths report we have used FDM for soluton of one dmensonal SWEs. 3.1 Intoducton to fnte dfference method (FDM) Fnte-dfference methods are numercal methods for approxmatng the solutons to dfferental equatons usng fnte dfference equatons to approxmate dervatves. From Taylors Seres expanson, we have Φ(x + x) = Φ(x) + Φ x ( x) Φ 2! x 2 ( x)2 +,..., + 1 n Φ n! x n ( x)n. (4) The grd generaton s well known n FDM and so has been depcted from Fg. 2 Fgure 2: Grd generaton n FDM In vew of Fg. 2 and Eq. (4) we can wrte ( ) ( ) Φ = Φ 2 +1 ( + Φ x + (1/2!)( x) 2 3 Φ + x x 2 x 3 Φ +1 ) (1/3!)( x) ( ) Φ x = (Φ+1 Φ ) x ( 2 Φ x 2 ) +1 ( 3 Φ (1/2!) x x 3 ) (1/3!)( x) 2. (5) As such forward dfference approxmaton may be wrtten as ( ) ( Φ Φ +1 = ) Φ + ( x) (6) x x Smlarly we have, backward dfference approxmaton as ( ) Φ = x ( Φ ) Φ 1 x 5 + ( x) (7)

12 Fnally the central dfference approxmaton s Φ +1 and ( ) Φ = Φ + x + x ( ) Φ Φ 1 = Φ x + x ( 2 Φ x 2 ( 2 Φ x 2 Subtractng (9) from (8), we get ) +1 ) +1 ( (1/2!)( x) 2 3 Φ + x 3 ( (1/2!)( x) 2 3 Φ x 3 ( ) Φ Φ +1 Φ 1 = 2 x + ( x) 2 x ( ) Φ = x 3.2 Dfferent schemes of FDM ) ) (1/3!)( x) (8) (1/3!)( x) (9) ( Φ +1 ) Φ 1 + ( x) 2 (10) 2 x We have used dfferent schemes of FDM such as Explct, Sem-mplct, and Implct schemes for numercal solutons of one dmensonal shallow water equatons. 3.3 Explct scheme To solve the SWEs n one dmenson we dscretze the frst Eq. (2) wth respect to both space and tme. We approxmate the tme dervatve by forward dfference and space dervatve by central dfference, thus we have [ η +1 η t [ η +1 η t + M x = 0 [ M +1 + M 1 = 0 2 x η t [ = M +1 2 x M 1 η +1 = η (c/2) [ M +1 M 1 (11) Also from second Eq. of (2), we have M t + gd η x = 0 6

13 M +1 [ [ M +1 M η +1 + gd η 1 = 0 t 2 x = M (1/2)c g (η + h) [ η +1 η 1 (12) Intally we are takng the basn depth to be zero..e. h = 0. So from the above the equaton we get M +1 = M (1/2)c g (η ) [ η +1 η 1 where, c = t, whch s the rato between tme step and spatal step. x 3.4 Sem-mplct scheme Usng ths scheme of FDM we can approxmate [4 the frst Eq. gven n (2) as follows [ η +1 [ η +1 η t + M x = 0 [ η 1 M +1 + M 1 = 0 t 2 x η t [ = M +1 2 x M 1 (13) η +1 = η (c/2) ( M +1 M 1 Now applyng Crank-Ncolson approxmaton to the Eq. (14) we have ) (14) ( M +1 M ) [ 1 = 1/2 (M +1 M +1 1 ) + (M+1 M +1 1 ) η +1 = η (c/4)[m +1 M +1 1 (c/4)[m+1 M +1 1 (15) Agan from the second Eq. of (2), we have [ M +1 M t M +1 M +1 M t + g D η x = 0 + g D [ η +1 η 1 2 x = 0 = M (1/2)c g [ η + h [ η +1 η 1 = M (1/2)c g [ η [ η +1 η 1 (16) 7

14 3.5 Implct scheme We now mplement another scheme of FDM known as mplct method [4on the one dmensonal SWEs. In ths method we have used Crank-Ncolson approxmaton, whch s the average of the central dfferences about the pont (, ) and (, + 1). Ths method has a stable soluton for any value of t and x. So, after applyng ths method n the frst Eq. of (2) and solvng we get η t + M x = 0 η+1 ( η +1 η + M +1 M 1 = 0 t 2 x ) η t = 2 x η +1 ( M +1 M ) 1 = η (c/2) ( M +1 M 1 Agan applyng Crank-Ncolson approxmaton to the above Eq. we have ) (17) ( M +1 M ) [ 1 = 1/2 (M +1 M +1 1 ) + (M+1 M +1 1 ) η +1 = η (c/4)(m +1 M +1 1 ) (c/4)(m+1 M +1 1 ) η +1 + (c/4)(m M +1 1 ) = η (c/4)(m +1 M 1 ) (18) From the second Eq. of (2) now we have [ M +1 M t M t + g D η x = 0 + gd [ η +1 η 1 2 x M +1 = M (1/2)cg ( η + h) ( η +1 η 1 M +1 +(1/4)cgh [ η η+1 1 = M (1/2)cg ( ) [ η η +1 [ η 1 (1/4)cgh η (19) Left hand sdes of Eqs. (18) and (19) are the terms of η and M at tme step + 1. The rght hand sdes are at tme step. So, we can solve the above system of Eqs. by any well known method. 3.6 Tsunam wave approxmaton ) = 0 +1 η 1 In prevous sectons, we have assumed the basn depth of water to be zero..e. h = 0. Now we have approxmated the basn depth h by a hyperbolc tangent [4 as gven below h(x) = tanh [ (x 70) where, 0 m x 100 m(20) 8 8

15 So, from Eq. (20) we can fnd the maxmum and mnmum value of h as 95 m and 5 m respectvely. Thus, for the explct scheme case.e. from Eq. (13) we have M +1 = M (1/2)c g (η ) [ η +1 η 1 M +1 = M (1/2)c g (η + h) [ η +1 η 1 (21) and for the mplct scheme case.e. from Eq. (19) we have M +1 M +1 M +1 = M (1/2)cg ( η + h) ( η +1 ) η 1 = M (1/2)c g ( ) ( η η +1 ) ( η 1 1/2c g h η +1 ) η 1 = M (1/2)c g ( ) ( η η +1 ) [ η 1 1/4c g h (η +1 η 1 ) + (η+1 +1 ) η+1 1 M +1 +1/4c g h ( η ) η+1 1 = M (1/2)c g ( ) ( η η +1 ) ( η 1 1/4c g h η +1 1) η (22) 4 Fnte volume method (FVM) for solvng one dmensonal SWEs In ths report we have used another numercal method known as fnte volume method (FVM) for soluton of one dmensonal SWEs. 4.1 Introducton to fnte volume method (FVM) FVM s a method of representng the partal dfferental equatons nto algebrac equatons. In ths method, we calculate the values at dscrete places or ponts as n fnte dfference or fnte element methods. Fnte (control) refers here as small volume surroundng each node pont on a mesh. In ths method, we ntegrate the gven partal dfferental equatons that contans a dvergence term whch can be converted to surface ntegral usng dvergence theorem. FVM s based on dscretzaton of the ntegral forms of the conservaton equatons. Dscretzaton s appled drectly to the ntegral equatons for small control (fnte) volumes as shown n the Fg. 3 [7 9

16 Fgure 3: One dmensonal control (fnte) volume n FVM In ths method, nstead of dscretzng frst, we start wth the ntegral form of the equatons. Below we wrte the steps nvolved n FVM for solvng one dmensonal SWEs. Step 1-Grd generaton n FVM: The frst step n the fnte volume method s grd generaton.e. by dvdng the doman nto dscrete control volumes. The boundares of control volumes are postoned md-way between adacent nodes. Thus each node s surrounded by a control volume or cell. Generally, t s better to set up control volumes near the edge of the doman n such a way that the physcal boundares concde wth the control volume boundares. Consder a control volume whose nodal pont s P and the neghbourng the nodes to the west and east, are defned as W and E respectvely. The west face of the control volume s referred by w and east face by e. The dstance from W to P, and P to E are gven by δx W P and δx P E respectvely as shown n the Fg. 4. Also the dstance from w to P, P to e are gven by δx wp and δx P e respectvely. The wdth of control volume [8 from w to e s denoted as x = δx we as shown n the fg (4). Fgure 4: One dmensonal grd for FVM [8 Step 2-Dscretzaton Concept: 10

17 The key step of FVM s the ntegraton of the governng equaton over a fnte ( control ) volume to yeld a dscretsed equaton at ts nodal pont P. Step 3- Soluton of the problem: Dscretsed equaton must be set at each of the nodal ponts n order to solve a problem.the resultng system of lnear algebrac equatons s then solved by any well known numercal method. 4.2 Dfferent schemes of FVM We have used two schemes of FVM namely upwnd nterpolaton (UI) and central dfference (CD) nterpolaton method for solvng one dmensonal shallow water equatons. 4.3 Upwnd nterpolaton It s the smplest way for approxmatng the partal dfferental equatons by FVM. Here, we use the value at a neghborng grd pont. We have taken the nearest upwnd ( upstream ) grd pont. Thus, takng the ntegral form of frst Eq. of (2) nto small control ( fnte ) volumes, and then dscretzed n the nearest grd pont. The volume ntegral s converted nto surface ntegral usng dvergence theorem. Thus, from frst Eq. (2) we have Integratng over control (fnte) volume, we get d dt x+1/2 Now we approxmate the above Eq. [7 η t + M x = 0 (23) x 1/2 ηdx + M e M w (24) d dt x+1/2 x 1/2 ηdx η P x (25) At the cell boundary.e. n the east face e = x +1/2, the normal n s n the postve drecton, so M e M P Agan at the cell boundary.e. at the west face w = x 1/2 the normal s n the negatve drecton. So, takng the value at the nearest grd pont n the west of the cell we have M w M W 11

18 So the fnte volume approxmaton of the above Eq. (24) s d dt (η P x) + M P M W = 0 [ [ η +1 η M + M 1 = 0 t x x ( η +1 η +1 η ) ( = t M M ) 1 = η c [ M M 1 where, c = t, whch s the rato of tme step and spatal step. x Agan from the second Eq. of (2) we have M t Integratng over control (fnte) volume, we get d dt x+1/2 x 1/2 Mdx + (26) + gd η x = 0 (27) x+1/2 x 1/2 g(η + h) η dx = 0 (28) t where, g s the acceleraton due to gravty, and h s the basn depth of water. Takng h to be zero, the Eq. becomes d dt x+1/2 x 1/2 Mdx + x+1/2 x 1/2 g(η) η t dx = 0 d dt x+1/2 x 1/2 Mdx + g x+1/2 x 1/2 η η x dx = 0 d dt (M P x) + g 2 [(η e + η w ) (η e η w ) = 0 [ M +1 M + g [ (ηp + η w )(η p η w ) = 0 t 2 x [ [ M +1 M + g (η + η 1 )(η η 1 ) = 0 t 2 x [ M +1 [ M +1 M t g = x 2 M c g = 2 [ (η + η 1 )(η η 1 ) = 0 [ (η + η 1 )(η η 1 ) = 0 where, c = t x M +1 = M cg 2 [ (η + η 1 )(η η 1 ) (29) 12

19 4.4 Central dfference (CD) nterpolaton Ths approxmaton s based on central dfference nterpolaton between two neghborng grd ponts. Usng ths nterpolaton we solve the frst Eq. of (2) n the followng way Integratng over control (fnte) volume of the grd ponts as shown n Fg. 4 we have d dt x+1/2 x 1/2 ηdx + x+1/2 x 1/2 M dx = 0 (30) x Approxmatng we have [7 d x+1/2 ηdx η P x (31) dt x 1/2 Now, the x dervatves are approxmated by central dfference nterpolaton. Accordngly we have the followng [ η +1 η x + (M) e (M) w = 0 t [ η +1 η x + [(M) E (M) W = 0 t [ η +1 η [ (M)E (M) W + = 0 t x [ [ η +1 η M +1 + M 1 = 0 t x η +1 = η c [ M +1 M 1 Agan from the second Eq. of (2) we have (32) M t + gd η x = 0 (33) Integratng over control (fnte) volume, we get d dt x+1/2 x 1/2 Mdx + x+1/2 x 1/2 Assume h to be zero, the Eq. becomes g(η + h) η dx = 0 (34) x d x+1/2 Mdx + g dt x 1/2 x+1/2 x 1/2 η η x = 0 (35) x 13

20 Also, d x+1/2 Mdx M P x (36) dt x 1/2 The x dervatves are approxmated by central dfference nterpolaton. Thus, we have the followng Eqs. as d dt [M P ( x) + g I = 0 (37) where, I = x+1/2 x 1/2 η η dx (38) x Now, solvng I near the neghbourng grd ponts we have I = 1 2 [(η E + η W )(η E η W ) I = 1 [ (η η 1 )(η +1 η 1 ) (39) Puttng the value of I n the Eq. (37) we get d dt (M P x) + g 1 2 [ M +1 M x + g t 2 M +1 M + t x M +1 = M cg 2 [ (η +1 + η 1 )(η +1 η 1 ) = 0 [ (η +1 + η 1 )(η +1 η 1 ) = 0 g [ (η η 1 )(η +1 η 1 ) = 0 [( η +1 + ) ( η 1 η +1 ) η 1 The values of M n the left hand sde of Eq. (40) represents the values at tme step + 1 and the rght hand sde term of Eq. represents the values at tme step. Thus we can solve Eqs. (32) and (40) for evaluatng the values of M and η by an teratve method. 4.5 Tsunam wave approxmaton Intally, n FVM we have assumed the basn depth of water to be zero..e. h = 0. Now we have approxmated the basn depth h by a hyperbolc tangent (as taken (40) 14

21 n FDM) h(x) = tanh [ Thus, from Eq. (28) we have (x 70) where, 0 m x 100 m(41) 8 d dt x+1/2 x 1/2 Mdx + g x+1/2 x 1/2 (η + h) η dx = 0 (42) x Let us assume, I = I = x+1/2 x 1/2 x+1/2 x 1/2 (η + h) η x η η x + x+1/2 x 1/2 h η x (43) Now, x+1/2 x 1/2 η η x = 1 2 [(η e + η w ) (η e η w ) (44) and, x+1/2 x 1/2 h η x = h (η e η w ) (45) So, I = 1 2 [(η e + η w ) (η e η w ) + h (η e η w ) (46) Now, puttng the value of I n Eq. (42) and solvng we get, d dt (M P x) + g 2 [(η e + η w ) (η e η w ) + gh (η e η w ) = 0 [ M +1 M + g t 2 x [(η p + η w )(η p η w ) + gh(η p η w ) = 0 [ M +1 M cg [ + (η 2 + η 1 )(η η 1 ) + gh [ η 1 η = 0 M +1 = M cg [ (η 2 + η 1 )(η η 1 ) gh [ η 1 η (47) 15

22 4.6 Interval fnte volume method (IFVM) The uncertan values occurred n practcal cases (such as errors n expermental data, and partal or mperfect knowledge of the parameters) may be handled by takng the uncertanty as nterval sense. So to compute these uncertantes we need nterval arthmetc. Let us consder the uncertan values of a parameter η n nterval form and the same may be wrtten n the followng way [ x, x = [ x x R, x x x (48) where, [ x and [ y are lower and upper values of the nterval respectvely. Let us assume two ntervals [ x, x and [ y, y then we have [ x, x + [ y, y = [ x + y, x + y [ x, x [ y, y = [ x y, x y (49) Upwnd Interpolaton IFVM: Now applyng the IFVM n Eqs. (26) and (29) we get [ M, M +1 = [ M, M (cg/2) [[ [ η, η + [ η, η [ 1 [ η, η [ η, η 1 [ η, η +1 = [ η, η c[ M, M [ M, M 1 (50) Rearrangng the above equatons we get a set of four Eqs. as follows [ M +1 [ M +1 = [ M (cg/2) [ η + [ η 1 [ η [ η 1 = [ M (cg/2) [ η + [ η 1 [ η [ η 1 [ η +1 = [ η c [ M [ M 1 [ η +1 = [ η c [ M [ M 1 (51) Central Dfference Interpolaton IFVM: Applyng the IFVM the Eqs. (32) and (40) we have [ M, M +1 = [ M, M (cg/2) [[ [ η, η +1 + [ η, η [ 1 [ η, η +1 [ η, η 1 [ η, η +1 = [ η, η c [[ M, M +1 [ M, M 1 (52) Rearrangng the above Eqs. one can get a set of four Eqs. as below [ M +1 [ M +1 = [ M (cg/2) [[ [ η +1 + [ η [ 1 [ η +1 [ η 1 = [ M (cg/2) [[ [ η +1 + [ η [ 1 [ η +1 [ η 1 [ η +1 = [ M c [ [ M +1 [ M 1 [ η +1 = [ M c [ [ M +1 [ M 1 (53) 16

23 5 Numercal results As mentoned earler we have used fnte dfference method ( FDM ) and fnte volume method ( FVM ) for soluton of one dmensonal SWEs. Moreover nterval fnte volume method (IFVM) has also been developed to solve the SWEs n uncertan envronment. 5.1 Numercal results usng FDM For one dmensonal shallow water equatons we have shown the ( numercal ) results 1 for dfferent schemes of FDM. We have used [4 grd sze x = m for space 8 and t = ( ) s for tme step. The boundary condtons [4 have been taken as at x = 0 and x = L M = 0. Also η(0, ) = η(0, ) ; η(l 1, ) = η(l, ) The ntal condtons are taken as at t = 0, assumng the ntal velocty of water s zero,.e. water s at rest poston at t = 0 and therefore M = 0. Frst we take the basn depth of water to be zero.e. h = 0. Another assumpton s, the maxmum and mnmum values of η are 18 m and 20 m respectvely. And the correspondng MATLAB program has been developed to compute the result for the behavour water surface elevaton (.e. η) wth dstance x s shown n Fg. 5 at (h = 0, t = 0) Fgure 5: At t = 0s, graph of η wth dstance x We have shown the graphs for varaton of water surface elevaton η wth dfferent value of tme t n case of explct scheme of FDM n Fg.6 to 10 for dfferent value of t. 17

24 Fgure 6: At tme t = 0.1 s Fgure 7: At tme t = s Fgure 8: At tme t = s Fgure 9: At tme t = 1 s Fgure 10: At tme t = s Agan we have shown the plots for varaton of water surface elevaton η wth dfferent values of tme t n case of mplct scheme of FDM n Fg.11 to

25 Fgure 11: At tme t = 0.1 s Fgure 12: At tme t = s Fgure 13: At tme t = s Fgure 14: At tme t = 1 s Fgure 15: At tme t = s Intally we have assumed the basn depth of water.e. h = 0. Now from Eq. (20) we take the mnmum and maxmum value of basn depth as h = 5 m and h = 95 m respectvely. Accordngly we have shown the behavour of η wth dstance n the Fg. 16 and 17 for both the cases. The behavour of water surface elevaton η wth dstance x when the basn depths are 85m 90m and 95m s depcted n Fg

26 Fgure 16: At t = 0.1s, graph of η wth dstance x when h = 5m Fgure 17: At t = 0.1s, graph of η wth dstance x when h = 95m Fgure 18: graph for h = 85m, 90m, 95m 5.2 Numercal results usng FVM and IFVM For one dmensonal shallow water equatons ( we ) have shown the numercal ( results) for 1 1 dfferent schemes of FVM. Grd sze x = m and tme step sze t = s have been consdered. The boundary condtons [4 have been taken smlar to FDM as at x = 0 and x = L M = 0. and η(0, ) = η(0, ) ; η(l 1, ) = η(l, ) The ntal condton has been taken as at t = 0, assumng the ntal velocty of water s zero,.e water s at rest poston at t = 0 and therefore M = 0 Smlar to the case of FDM we frst take the basn depth of water to be zero.e the value of h = 0. Another assumpton s the maxmum and mnmum values of η are 18 m and 20 m respectvely. Thus n case of FVM when the ntal velocty of water s zero.e at t = 0 the varaton of water surface elevaton η wth dfferent value of x s shown n Fg. 19 usng MATLAB. 20

27 Fgure 19: At t = 0s, graph of η wth dstance x The plots for varaton of water surface elevaton η wth dfferent value of tme t n case of upwnd nterpolaton of FVM has been shown n Fg. 20 and 21. Fgure 20: At tme t = s Fgure 21: At tme t = 0.5 s Smlarly the plots for varaton of water surface elevaton η wth dfferent values of tme t n case of central dfference nterpolaton of FVM has been shown n Fg. 22 to 27. Fgure 22: At tme t = 0.1 Fgure 23: At tme t =

28 Fgure 24: At tme t = Fgure 25: At tme t = 1 Fgure 26: At tme t = 1 Fgure 27: At tme t = As per Eq. (20) the mnmum and maxmum value of basn depth.e h = 5 m and h = 95 m respectvely have been consdered. The behavour of η wth dstance s cted n Fg. 28 and Fg 29. Fgure 28: At t = 0.1s, graph of η wthfgure 29: At t = 0.1s, graph of η wth dstance x when h = 5m dstance x when h = 95m 22

29 Fgure 30: graph of η wth dstance x Numercal results usng IFVM In ths case the values of η and M, are n an nterval as such the maxmum and mnmum values of η are taken n the nterval as [ and [ respectvely. Also, the value of M n the nterval has been as [0, 0.5. After takng the above values n the nterval we have nvestgated the varaton of water surface elevaton η wth dstance x wth upper nterval, lower nterval and crsp value whch s shown n Fg Comparson between FDM and FVM, concluson and future drectons A comparatve study of numercal results of FDM and FVM for one dmensonal shallow water equatons have been presented hare. After nvestgatng the numercal methods FDM and FVM for soluton of one dmensonal shallow water equaton we have compared the values for water surface elevaton η wth dstance x and s shown n the Fg. 31 and 32. Correspondng comparson tables for both the methods wth dfferent tme are gven n Tables 1 and 2. 23

30 Table 1: At tme t = 0.1 comparson of FDM wth FVM for the value of water surface elevaton S.no FDM FVM Table 2: At tme t = 0.1 for FVM and at tme t =.1667 for FDM the value of water surface elevaton S.no FDM FVM

31 Fgure 31: At tme t = 0.1 s Fgure 32: At tme t = 0.1 s for FVM and t = s for FDM Concluson Although the results for one dmensonal SWEs usng fnte volume schemes concde wth the smple schemes of fnte dfference but advantage of FVM s that of the meshng scheme. The unque character of fnte volume schemes usually appear when multdmensonal problems are solved usng unstructured grds. The most mportant feature of FVM s that the method s conservatve because the flux enterng a gven volume s dentcal to that leavng the adacent volume and t s used only when the equatons are based on conservaton of physcal laws. Another advantage of the FVM s that t s easly formulated to allow for unstructured meshes. The method s wdely used n computatonal flud dynamcs (CFD). Also, n FDM the values of the dependent varables are stored at the nodes only. In FEM these values are stored at the element nodes. But n FVM, the values of the dependent varables are stored at the centre of the control volume. In case of FDM and FEM, conservaton of mass, momentum, energy are not ensured at each cell/control volume. But ths s true for FVM. It may be worth mentonng that FVM takesless tme n computaton because t converges wth less number of control volumes. Future Drecton Ths nvestgaton gves a new dea of the Interval FVM through SWEs and ths can very well be used n future research for better results for other equatons obtaned from dfferent applcatons. The dea may easly be extended to other structured as well as unstructured problems wth varous complcatng effects. Although ths requre more complex forms of nterval computaton to handle the correspondng problem. 25

32 References [1 Imamura. F, Yalcne. A.C, Tsunam Modelng Manual (Draft), pp- (1-58), [2 George. F, Carrer. T, Harry. Y, Tsunam run-up and drawn-down on a plane beach. J. Flud Mech. pp-(79-99), [3 Goto. C, Ogawa. Y, Numercal Method of Tsunam Smulaton wth the Leap- Frog Scheme.IUGG/IOC Tme Proect, Manuals and Gudes. No.35, Pars, 4 parts, [4 Junbo. P, Harvey Mudd College, Numercal smulaton Of Wave Propagaton Usng the Shallow Water Equatons, Web: dyong/math164/2007/park/fnalreport.pdf [30th Aprl 2014 at 9:31pm, [5 Shukla. A, Sngh. A. K, Sngh. P, A Comparatve Study of Fnte Volume Method and Fnte Dfference Method for Convecton-Dffuson Problem. Amercan Journal of Computatonal and Appled Mathematcs ; 1(2): pp- (67-73) [6 Boussnesq. F, travaux de M de Sant-Venant,Annales des ponts et chausses 12, pp- ( ), [7 O. Zkanov. Essental Computatonal Flud Dynamcs. Wley Inda Pvt.Ltd pp- (86-102), [8 H. K. Versteeg, W. Malalasekera. An ntroducton to computatonal flud dynamcs The fnte volume method. Longman Scentfc and Techncal. pp- (85-133), [9 T. Cebec, J. R. Shao, F. Kafyeke, E. Laurendeau. Computatonal Flud Dynamcs for Engneers Horzon Publshng., pp- ( ), [10 C. Dawson, M. Mrabto. Insttute for Computatonal Engneerng and Scences,Unversty of Texas at Austn [11 R. B. Bhat, S. Chakraverty. Numercal Analyss n Engneerng, Narosa Publshng House Pvt. Ltd [12 M. K. Jan, S.R.K. Iyengar, R.K. Jan. Numercal Methods For Scentfc and Engneerng Computaton, New Age Internatonal Lmted, Publsher, [13 A. batra. Fnte Volume Method:basc prncples and examples,department of Mathematcs and Computng, IIT Guwahat,