Numerical techniques to simulate tsunamis are described. Partial differential equations are reduced to a system of ordinary differential equations to which appropriate numerical solvers can be applied. The techniques are illustrated with an example in which tsunami due to an earthquake is simulated

Liu, Ying

Wang, Ming Jun

Watanabe, Masaji

Okayama University Scientific Achievement Repository

Journal of the Faculty of Environmental Science and Technology,Okayama UniversityVoUl, No,l, pp,15·18, March 2006Numerical Techniques for Simulation of Tsunami Basedon Finite ElementsMasaji Watanabel , Ying Liu2 and Ming Jun Wang3(Received December 27, 2005)Numerical techniques to simulate tsunamis are described. Partial differential equations are re-duced to a system of ordinary differential equations to which appropriate numerical solvers can beapplied. The techniques are illustrated with an example in which tsunami due to an earthquakeis simulated.Key words: tsunami, finite elements, numerical simulation1 INTRODUCTION 2 REDUCTION OF TSUNAMI MODELwhere Xj and Yj are the coordinates of the jth node.(Pi(Xj,Yj)=bij, i=1,2, ... ,n, j=1,2, ... ,n,A mathematical model for simulation of tsunami is thesystem of partial differential equations:M= t;udz, N= [(vdz.J-h Lh(1)aM o(at + 9 (h +() 8x = 08N 8(tit +9 (h +() 8y = 08( + 8M + aN = 0at ax 8ySuppose that a region in xy plane is divided into mtriangular elements with n nodes. Let (Pi (x,y) be thecontinuous piecewise linear basis function that satisfiesThose equations are based on the continuity equationand the equations of motion to analyze long waves(Dean and Dalrymple, 1991). Here, z = (and z = -hcorrespond to the water surface and the ocean floor,respectively. Let u and v denote x-component and Y-component of the velocity, respectively. M and N aredefined byThe earthquake of magnitude 8.9 occurred in an off-shore area of the north end of Sumatra around 1 a.m.GMT on December 26,2004. The tsunami caused bythe earthquake reached the maximum height of 10 m,and devastated the coastal areas of Indian Ocean (De-cember 27, 2004, YOMIURI ONLINE, in Japanese).The number of the deaths due to the disaster exceeded230,000 (January 25, 2005, YOMIURI ONLINE, inJapanese). The establishment of warning systems toprevent damages from such disasters is essential.Techniques based on shallow water equations havebeen proposed to analyze tsunami waves (Kawaharaet al., 1988, Kodama et al., 1993, Lye, 2005) In thispaper, a technique based on finite elements is intro-duced to simulate tsunami waves. In the followingsections, techniques to analyze flows generated in Ko-jima Lake as well as numerical results are introduced.In Section 2, a technique to reduce partial differentialequations to a system of ordinary differential equationsare described. In Section 3, those techniques are illus-trated with an example. A tsunami in Nankai Trougharea is simulated. In Section 4, the difference betweenthe technique introduced in this paper and some othermethods is discussed.lGraduate School of Environmental Science, Okayama Uni-versity.2Department of Environmental and Mathematical Sciences,Faculty of Environmental Science and Technology, OkayamaUniversity.3Graduate School of Natural Science and Technology (Mas-ter's Course), Okayama University.1516 J. Fac. Environ. Sci. and Tech .. Okayama Univ.11 (1) 2006Then M, N, (, and h can be given bynM (x,y, t) :::::! I:Mj (t)¢j (x,y)j=lnN(x,y,t) :::::! I:Nj (t)¢j (x,y)j=ln((x,y,t) :::::! I: (j (t) ¢j (x, y)j=lnh (x, y) :::::! I: hj¢j (x, y)j=lSubstitution of these expressions into the system (1)leads toThe value of the partial derivatives on the right-hand sides at (x, y) = (Xi, Yi) can be approximatedby the average of the partial derivatives over the ele-ments which have the node in common as one of theirvertices. Then the system (2) can be regarded as asystem of ordinary differential equations.For (x, y) = (Xi, Yi), these equations becomedMi a(dt = -g (hi + (i) axdNi a(----;It = -g (hi + (i) ayd(i aM aN- -----dt ax 8y(2)Figure 1: Topography of the ocean floor.on this rectangular grid was set. It consists of 160000elements and 80601 nodes.Figure 1 shows the topography of the ocean floorgenerated by the values of h introduced into the com-putational analysis. The depth of the ocean in the arearanges approximately from 0 m to 4500 m. Figure 2shows the surface of the sea initially set. It is basedon Gaussian hump (Lye, 2005).(x - xO)2 + (y - YO)2 < 4(72(x - XO)2 + (y - YO)2 ~ 4(72Here (xo, Yo) corresponds to the east longitude 134°and the north latitude 32°, a = 10, (7 = 10000. Theinitial values of (i were generated according to thisexpression of (. The initial values of M i and N i3 NUMERICAL RESULTS FOR TSUNAMISIMULATIONA coordinate system whose origin corresponds to theeast longitude 130° and the north latitude 30° and unitlength corresponds to 1 meter was set. A tsunami wassimulated over a rectangular regionXmin ~ X ~ X max , Ymin ~ Y ~ Ymax,where Xmin = 200000, Xmax = 1000000, Ymin = 10000,and Ymax = 410000. The region lies between east lon-gitude 131° and 140° and north latitude 30° and 36°,and contains a part the north edge of Nankai 'Though.The interval [Xmin, x max ] was divided into 400 equallyspaced intervals. Similarly, the interval [Ymin, Ymax]was divided into 200 equally spaced intervals, and arectangular grid was set. A finite element mesh basedwere set.The fourth-order Adams-Bashforth-Moulton predictor-corrector (PECE mode) was implemented to solve thesystem (2) numerically in conjunction with the fourth-order Runge-Kutta method to generate approximatesolutions in the first three steps (Lambert, 1973). Fig-ures 3, 4, 5, and 6 show the transition of the surfaceof the sea.4 CONCLUSIONIt has been shown that partial differential equations offluid dynamics can be reduced to system of ordinarydifferential equations to simulate tsunami implement-ing a finite element mesh. It has also been shown thatODE solvers can be applied to solve those ordinaryt.lasaji WATANABE et al. I :\umerical Techniques for Simulation of Tsunami 17Figure 2: Initial surface of the sea.~.-.Figure 3: Surface of the sea at t = 60..... .-_. -.--.Figure 4: Surface of the sea at t = 300.Figure 5: Surface of the sea at t = 600.18 J. Fac. Em·iron. Sci. and Tech .. Okayama Univ.ll (1) 2006Figure 6: Surface of the sea at t = 900.differential equations effectively. The values of spatialderivatives at each node was replaced with average val-ues of the derivatives over the elements that have thenode in common as one of vertices (Okamoto et al.,1992) and the method of collocation (Mitchell et al.,1977) was implemented. The technique has an advan-tage over a method based on weak formulation thatleads to a system of linear equations with a stiffnessmatrix. It also has an advantage over a finite differ-ence method for it can be adapted to the structure ofa coastline.ACKNOWLEDGMENTSData concerning the depth of the ocean are based onthe Ocean Sounding Chart, G1406, produced by theHydrographic Department, M. S. A., Japan, 1985, andBathymetric Chart, 6602, Nippon, Offing of Tokai-Kii,compiled from the various sources of the HydrographicDepartment, M. S. A., Japan, to 1992, 1993, MaritimeSafety Agency, Tokyo. The articles of YOMIURl ON-LINE cited in this paper were featured in the web site,http://www.yomiuri.co.jp/features/india/ .REFERENCESDean, R. G. and Dalrymple R. A. (1991): Water WaveMechanics for Engineers and Scientists, AdvancedSeries on Ocean Engineering - Volume 2, WorldScientific, Singapore.Kawahara, M., Kodama, T., and Kinoshita, M. (2005):Finite element method for tsunami wave prop-agation analysis considering the open boundaryconditions, Computers & Mathematics with Ap-plications, 16 (1/2) (1998), 139-152.Kodama, T. and Kawahara, M. (1993): A finite el-ement analysis of shallow water flow with openboundary condition, Computational Mechanics,Valliappan, Pulmano & Tin-Loi (eds) © 1993Balkema, Rotterdam, ISBN 90 5410 333 7 (1993),11-18.Lambert, J. D. (1973): Computational Methods inOrdinary Differential Equations, John Wiley &Sons, Chichester.Lye, K. H. (2005): Modeling propagation of 2004Asian Tsunami: A theoretical Analysis, Inter-national Conference of Reservoir Operation andRiver Management, Guangzhou & Three Gorges,China, September 17-23, 2005.Mitchell, A. R. and Wait, R. (1977): The Finite El-ement Method in Partial Differential Equations,John Wiley & Sons, Chichester.Okamoto, T, Kawahara, M., loki, N., and Nagaoka,H. (1992): Two-dimensional wave run-up anal-ysis by selective lumping finite element method,International Journal for Numerical Methods inFluids, 14 (1992), 1219-1243.

Numerical Techniques for Simulation of Tsunami Based on Finite Elements

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