Tsunami Modeling1

Randall J. LeVeque

Department of Applied Mathematics, Universityof Washington, Seattle, WA 98195-2420.

1 Introduction

Appreciation for the danger of tsunamis amongthe general public has soared since the IndianOcean tsunami of 26 December 2004 killed morethan 200,000 people. Several other large tsunamishave occurred since then, including the devastat-ing 11 March 2011 Great Tohoku tsunami gen-erated off the coast of Japan. The internationalcommunity of tsunami scientists has also grownconsiderably since 2004 and an increasing num-ber of applied mathematicians have contributedto the development of better models and compu-tational tools for the study of tsunamis. In ad-dition to its importance in scientific studies andpublic safety, tsunami modeling also provides anexcellent case study to illustrate a variety of tech-niques from applied and computational mathe-matics. This article combines a brief overviewof tsunami science and hazard mitigation withdescriptions of some of these mathematical tech-niques, including an indication of some challeng-ing problems of ongoing research.

The term “tsunami” is generally used to re-fer to any large scale anomolous motion of waterthat propagates as a wave in a sizable body of wa-ter. Tsunamis differ from familiar surface wavesin several ways. Typically the fluid motion is notconfined to a thin layer of water near the surfaceas it is in wind-generated waves. Also the wave-length of the waves is much longer, sometimeshundreds of kilometers. This is orders of magni-tude larger than the depth of the ocean (which isabout 4 km on average) and so tsunamis are alsosometimes referred to as “long waves” in the sci-entific literature. In the past, tsunamis were oftencalled “tidal waves” in English because they sharesome characteristics with tides, which are the vis-ible effect of very long waves propagating aroundthe earth. However, tsunamis have nothing to do

1Draft of 13 November 2012. To appear in the Prince-ton Companion to Applied Mathematics (N. J. Higham,Editor)

with the gravitational (tidal) forcing that drivesthe tides, and so this term is misleading and isno longer used. The Japanese word “tsunami”means “harbor wave”, apparently because sailorswould sometimes return home to find their har-bor destroyed by mysterious waves they did notobserve while at sea. Strong currents and vor-tices in harbors often cause extensive damage toships and infrastructure even when there is noonshore inundation. Although the worst effectsof a tsunami are often observed in harbors, theeffects can be devastating in any coastal region.Because tsunamis have such a long wavelength,they frequently appear onshore as a flood thatcan continue flowing inward for tens of minutesor even hours before flowing back out. The flowvelocities can also be quite large, with the con-sequence that even a tsunami wave with an am-plitude of less than 1 meter can sweep people offtheir feet and do considerable damage to struc-tures. Tsunamis arising from large earthquakesoften result in flow depths greater than 1 meter,particularly along the coast closest to the earth-quake, where runup can reach 10s of meters.

Tsunamis are generated whenever a large massof water is rapidly displaced, either by the mo-tion of the seafloor due to an earthquake or sub-marine landslide, or when a solid mass entersthe water from a landslide, volcanic flow, or as-teroid impact. The largest tsunamis in recenthistory, such as the 2004 or 2011 events men-tioned above, were all generated by megathrustsubduction zone earthquakes at the boundary ofoceanic and continental plates. Offshore frommany continents there is a subduction zone whereplates are converging. The denser material in theoceanic plate subducts beneath the lighter conti-nental crust. Rather than sliding smoothly, stressbuilds up at the interface and is periodically re-leased when one plate suddenly slips several me-ters past the other, causing an earthquake duringwhich the seafloor is lifted up in some regions anddepressed in others. All of the water above theseafloor is lifted or falls along with it, creatinga disturbance on the sea surface that propagatesaway in the form of waves. See Figure 1 for an il-lustration of tsunami generation and Figure 2 fora numerical simulation of waves generated by the2011 Tohoku earthquake off the coast of Japan.

1

2

This article primarily concerns tsunamis causedby subduction zone earthquakes since they are amajor concern in risk management and have beenwidely studied.

2 Mathematical models andequations of motion

Tsunamis are modeled by solving systems of par-tial differential equations (PDEs) arising fromthe theory of fluid dynamics. The motion ofwater can be very well modeled by the Navier-Stokes equations for an incompressible viscousfluid. However, these are rarely used directlyin tsunami modeling since they would have tobe solved in a time-varying three-dimensional do-main, bounded by a free surface at the top and bymoving boundaries at the edges of the ocean asthe wave inundates or retreats at the shoreline.Fortunately, for most tsunamis it is possible touse “depth-averaged” systems of PDEs, obtainedby integrating in the vertical z direction to obtainequations in two space dimensions (plus time).In these formulations, the depth of the fluid ateach point is modeled by a function h(x, y, t) thatvaries with location and time. The velocity ofthe fluid is described by two functions u(x, y, t)and v(x, y, t) that represent depth averaged val-ues of the velocity in the x- and y-directions re-spectively. In addition to a reduction from threeto two space dimensions, this eliminates the freesurface boundary in z; the location of the sea sur-face is now determined directly from the depthh(x, y, t). These equations are solved in a time-varying two-dimensional (x, y) domain since themoving boundaries at the shoreline must still bedealt with.

A variety of depth-averaged equations can bederived, depending on the assumptions madeabout the flow. For large-scale tsunamis, the so-called “shallow water” equations (also called theSt. Venant or long-wave equations) are frequentlyused and have been shown to be very accurate.The assumption with these equations is that thefluid depth is sufficiently shallow relative to thewavelength of the wave being studied. This isgenerally true for tsunamis generated by earth-quakes, where the wavelength is typically 10–100times greater than the ocean depth.

The two-dimensional shallow water equationshave the form

ht + (hu)x + (hv)y = 0,

(hu)t +

(hu2 +

1

2gh2

)x

+ (huv)y = −ghBx,

(hv)t + (huv)x +

(hv2 +

1

2gh2

)y

= −ghBy,

(1)

where subscripts denote partial derivatives, e.g.ht = ∂h/∂t. In addition to the variables h, u, valready introduced, which are each functionsof (x, y, t), these equations involve the gravita-tional force g and the topography or seafloorbathymetry (underwater topography) denoted byB(x, y). Typically B = 0 represents sea levelwhile B > 0 is onshore topography and B < 0represents seafloor bathymetry. Water is presentwherever h > 0 and η(x, y, t) = h(x, y, t)+B(x, y)is the elevation of the water surface. See Fig-ure 3 for a diagram in one space dimension. Dur-ing an earthquake B should also be a functionof t in the region where the seafloor is deform-ing. In practice it is often sufficient to includethis deformation in B(x, y) while the initial con-ditions for the depth h(x, y, 0) are based on theundeformed topography. The seafloor deforma-tion then appears instantaneously in the initialsurface η(x, y, 0), which initializes the tsunami.In the remainder of this article, the term topog-raphy will be used for both B > 0 and B < 0 forsimplicity.

If B(x, y) < 0 is constant (a flat bottom) thenthe “source terms” on the right hand side of theseequations drop out and the equations model theconservation of mass (h) and momentum (hu, hv).Over a varying bottom, mass is still conserved butmomentum is affected by the terrain, as seen forexample in the reflection of waves at a shorelineand partial reflection when a wave interacts withunderwater features. The term 1

2gh2 appearing in

the momentum equations is the depth averaged“hydrostatic pressure” in a column of water ofdepth h. (This and all other terms in (1) shouldin fact also involve the fluid density ρ, but thiscancels out everywhere if the density is assumedto be constant.)

The equations (1) are a nonlinear system ofequations of hyperbolic type. Hyperbolic PDEs

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Figure 1: Illustration of the generation of a tsunami by a subduction zone earthquake.

frequently arise when waves are modeled math-ematically, as described in [article on hyperbolicPDEs]. The amplitude of a tsunami in the deepocean is generally very small relative to the waterdepth; typically less than a meter even for a largemegathrust tsunami. Away from the coast theseequations could be approximated by linearizedequations with variable coefficients arising fromthe varying topography. Near shore, however,the amplitude of the wave is large relative to thedepth of the fluid and the full nonlinear equationsmust be used to accurately model the interactionof a tsunami with the nearshore topography andthe onshore inundation that occurs. Solutions tononlinear hyperbolic PDEs can become discon-tinuous if a shock develops. In the case of theshallow water equations, a shock is also called a“hydraulic jump” and is a mathematical ideal-ization of a thin region in which the depth andvelocity of the fluid jumps from one value to an-other. Such regions frequently appear as a tur-bulent wave front (sometimes called a “turbulentbore”) once the tsunami moves into sufficientlyshallow water. The shallow water equations donot model this turbulent zone directly, but arefrequently adequate to capture important quan-tities such as the depth and fluid velocities behindthe bore and its propagation speed.

3 Uses of tsunami modeling

The PDEs describing a tsunami cannot be solvedexactly, in general, and so numerical methods

must be used to simulate the propagation andinundation of a tsunami. A brief description ofhow this might be done and some of the chal-lenges that arise is given in Section 4, but firstwe motivate the need for numerical models bydescribing some common uses of such models.

3.1 Real-time warning systems

One natural use of a numerical model is to assistin issuing warnings in real time as a tsunami prop-agates across the ocean, and to determine whatcoastal regions should be evacuated. There aremany challenges to doing this quickly and accu-rately. Accurate assessment is critical not only toinsure that areas at risk are properly warned butalso to avoid triggering evacuation in areas whereit is not necessary, which can itself cause loss oflife, serious financial impact, and decreased atten-tion to future warnings. For a subduction zonemegathrust earthquake it is often impossible toissue tsunami warnings quickly enough for areasalong the nearby coastline. The tsunami may ar-rive in less than an hour, often sooner, and itis critical that residents understand the need tomove to high ground when a major earthquakeoccurs. On the other hand, across the ocean theearthquake itself is not felt and so provides nodirect indication of an impending tsunami, butthere are several hours available to perform sim-ulations and issue warnings.

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140 144 148Longitude

34

38

42

46La

titu

de

20 minutes after earthquake

140 144 148Longitude

34

38

42

46

Lati

tude

30 minutes after earthquake

140 150 160Longitude

30

40

50

Lati

tude

1 hour after earthquake

140 150 160Longitude

30

40

50La

titu

de

2 hours after earthquake

Figure 2: Propagation of the tsunami arising from the 11 March 2011 Tohoku earthquake off the coast of Japan

at four different times from 20 minutes to 2 hours after the earthquake. Waves propagate away from the source

region with a velocity that varies with the local depth of the ocean. Contour lines show sea surface elevation

above sea level, in increments of 20 cm (top, at early times) and 10 cm (bottom, at later times). There is a

wave trough behind the leading wave peak shown here, but for clarity the contours of elevation below sea level

are not shown.

3.2 Tsunami source inversion

To perform tsunami simulations, it is necessary toestimate the source, i.e., the deformation of thesea floor that generates the tsunami, since thisdetermines the initial conditions that are usedto numerically solve the PDEs modeling tsunamipropagation and inundation. There is generallyno way to measure this directly, and so someform of inverse problem [pointer to article on in-verse problems?] must be solved to obtain an

estimate of the deformation based on measure-ments that can be made, such as the earth mo-tion due to seismic waves caused by the earth-quake, or measurements of the tsunami itself. Ini-tial estimates of the location and magnitude of anearthquake generally come from analyzing record-ings of seismic waves, which are compression andshear waves that travel through the earth withmuch higher velocity than tsunamis and that areroutinely recorded at hundreds of seismometerswidely scattered around the world. From the

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Figure 3: Illustration showing the notation used in the shallow water equations (1).

measured wave forms at many locations it is pos-sible to construct an estimate of how the earthmust have moved to produce this set of data. Thisrelies ultimately on solving an inverse problem forthe PDEs modeling wave motion in elastic mate-rials [pointer to article on elasticity? Any articleon seismology?]. Seismic inversions generally es-timate the slip of the earth along the earthquakefault, which may be 10s of kilometers below thesea floor. Converting this slip on the fault planeto deformation of the sea floor requires solving an-other elasticity problem, whose solution is oftenapproximated by the Okada model. This is basedon the Green’s function for the deformation ofthe boundary of an elastic half space caused by adelta function dislocation. Integrating this overa finite-sized patch of a fault plane gives an esti-mate of the resulting seafloor displacement.

While the results of seismic inversions are in-valuable in modeling tsunamis, performing an ac-curate inversion requires collecting and process-ing a large amount of data and this may not befeasible in real time. In order to gather betterinformation about tsunamis as they propagate, anumber of pressure gauges have recently been de-ployed on the seafloor that are able to measurewater pressure extremely accurately. From thehydrostatic pressure it is possible to estimate thedepth of the water at these locations with enoughprecision to capture variations due to a tsunamipassing by. Direct measurement of a tsunami atone or more of these gauges can then be combinedwith seismic models identifying the approximatesource location and geophysical knowledge of thefaults that are most likely to produce tsunamis.

This information, together with accurate tsunamipropagation models, can allow the solution of aninverse problem in order to estimate the seafloordeformation that caused the tsunami more accu-rately and quickly than is possible using seismicinformation alone.

3.3 Hazard modeling and mitigation

Real-time simulations of tsunamis are used to is-sue warnings, but tsunami modeling has many on-going uses beyond this. Protecting communitiesrequires adequate planning long before a tsunamitakes place, and tsunami models are used to simu-late the effect of tsunamis arising from hypotheti-cal earthquake events. The results of such modelscan be used to determine what regions of a com-munity are most at risk and what regions can bedesignated as safe zones for evacuation. Model-ing the arrival time and pattern of the waves canbe used in connection with traffic-flow models ofevacuation. Some communities in tsunami-proneregions build sea walls or gates that can be closedfor protection against tsunamis, or build “verti-cal evacuation structures” in regions where thereis no easily accessible high ground for large-scaleevacuation. These structures may take the formof multi-use buildings built to withstand tsunamisand tall enough that the upper floors are safehavens, or may consist of large berms that formartificial high ground. Designing such structuresrequires modeling the flow depth and often alsothe fluid velocities of hypothetical tsunamis.

Of course it is impossible to know exactly whatthe seafloor deformation will be for future earth-

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quakes, but quite a bit is known about the ma-jor subduction zones and the likely location andmagnitude of large earthquakes based on the ge-ology and past history. There is always a ques-tion of how large a tsunami one should designfor. Sometimes an estimate of the “credible worstcase” tsunami for that location is used, but thismay correspond to an event with very low prob-ability of occurence that would require an enor-mous expenditure to protect against, money thatmight be better spent protecting against morelikely events at additional locations. To bet-ter understand these tradeoffs, recently there hasbeen increased interest in probabilistic tsunamihazard assessment (PTHA) in which a set of pos-sible events are assigned probabilities, or an en-tire spectrum of possible events is assigned someprobability density function, typically over a veryhigh-dimensional stochastic space. The goal isthen to obtain from this a probabilistic descrip-tion of the resulting inundation patterns, flowdepths, velocities, etc. This is a form of uncer-tainty quantification (UQ), a rapidly growing fieldof importance in many fields of computational sci-ence where simulations are based on many uncer-tain inputs and the goal is a probabilistic descrip-tion of the resulting outputs rather than a singlesimulation result. Applied mathematicians andstatisticians have a large role to play in the de-velopment of new techniques to efficiently solvethese problems. [Pointer to article on UQ?]

3.4 The study of past tsunamis andearthquakes

Another major use of tsunami modeling is thestudy of past tsunamis. A wealth of data hasbeen collected following recent tsunami events by“tsunami survey teams” that measure inundationand runup along affected coasts. There is alsodata available from seafloor pressure gauges, tidegauges along the coast, and other data collectionfacilities. Models of the seafloor deformation pro-duced by solving the source inversion problem canthen be used as initial data for tsunami modelsand the computed results compared with mea-surements. Such studies are important in veri-fying that a tsunami model gives a sufficientlyaccurate approximation to a real tsunami that itcan be used with confidence for warning or haz-

ard mitigation purposes. [Pointer to article onVerification and Validation?] Validated modelsare also used in performing tsunami source inver-sion to estimate the seafloor deformation, and cangive additional insight into the earthquake mech-anism that is useful to seismologists. Tsunamimodels can also help explain unusual features ofpast events by providing a laboratory for explor-ing the fluid dynamics taking place during theevent.

Tsunami models can also help reconstructevents that happened in the more distant past,for which there are no pressure gauge or tidegauge data and perhaps limited historical recordsof the regions inundated, or no human records inthe case of prehistoric events or those that oc-curred on uninhabited coastlines. Luckily, formany events a geological record of the tsunamiinundation is recorded in the form of tsunami de-posits. As a tsunami approaches shore it typi-cally becomes turbulent and picks up sedimentfrom the seafloor, such as sand and marine micro-organisms. This material is carried inland duringthe flooding stage and typically settles out of theflow as the flow decelerates and reverses, leavingbehind a layer of deposits, often far inland. Intsunami-prone areas there are often many layersof tsunami deposits that have been built up overthousands of years, separated by layers of soil thatslowly build up between tsunamis. Core samplesor trenches can reveal many past events that canoften be dated using radiocarbon dating of or-ganic matter or interspersed tephra layers fromknown volcanic eruptions. The study of tsunamideposits is a major source of information aboutthe magnitude, location, and recurrence timesof past earthquakes. This information is criti-cal in developing probabilistic models of tsunamior earthquake hazards, as well as to obtaining abetter scientific understanding of earthquake pro-cesses. Numerical tsunami models can be usedto help identify the location and magnitude ofseafloor deformation that would lead to the pat-terns of inundation recorded by tsunami deposits.Models that include sediment erosion, transport,and deposition are also being used to better un-derstand the fluid dynamics of the creation oftsunami deposits, which ultimately will lead tomore accurate descriptions of the tsunamis that

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caused observed deposits.

4 Numerical modeling

Systems of nonlinear PDEs such as the nonlinearshallow water equations (1) typically cannot besolved exactly except for very simple cases, for ex-ample a one-dimensional wave on a linear beach.Realistic tsunami modeling always relies on nu-merical solution of the PDEs. This requires dis-cretizing the equations in some manner: replac-ing the differential equations describing the con-tinuum solution (defined for all (x, y, t) in somedomain) by a finite set of discrete algebraic equa-tions whose solution can be computed in finitetime on a computer. There are many ways to dothis, and general discussions of numerical solu-tion of differential equations are given in [otherarticles??].

Finite difference methods are often used, inwhich a discrete grid is introduced consisting ofa finite number of grid points (xi, yj) coveringthe domain, and the solution is approximatedonly at these points at a discrete set of timest0, t1, t2, . . .. Derivatives in the PDE are re-placed by finite difference approximations basedon the approximate solution at neighboring gridpoints, obtaining a discrete set of algebraic equa-tions that can be solved on a computer. An-other popular approach is to use a finite volumemethod, in which the domain is subdivided intoa finite number of grid cells and the approximatesolution consists of average values of the solutionover each grid cell. Integrating the PDEs over agrid cell gives an expression for the time deriva-tive of the cell average that can be used to up-date the cell averages from one time tn to thenext time tn+1. To obtain better accuracy, meth-ods are sometimes used in which the solution oneach grid cell is approximated by a polynomialrather than by only the cell average (which canbe interpreted as a constant function, or polyno-mial of degree 0, over each cell). In this case,the higher order coefficients of each polynomialmust be updated from one timestep to the next.A method of this type that has recently becomepopular for tsunami modeling is the Discontinu-ous Galerkin method, in which the piecewise poly-nomial function obtained from the polynomials

defined on each cell is not assumed to be con-tinuous at the interface between one cell and itsneighbor. The term “Galerkin” refers to a finiteelement approach to deriving equations for evolv-ing the polynomial coefficients in time. [Pointersto other sections.]

4.1 Nonlinearity and shock formation

A prominent feature of nonlinear hyperbolicPDEs is that shocks can form in the solutions,which are discontinuities in the depth and veloc-ity that can arise even from smooth initial con-ditions. [Article on shocks or nonlinear conserva-tion laws?] As mentioned in Section 2, these cor-respond to hydraulic jumps or bores that are seenin tsunamis as they approach the shore. Sharpdiscontinuities are only an approximation to thetrue behavior, but often give a good approxima-tion to the flow. Incorporating more accuratefluid dynamics models would lead to systems ofPDEs that are much more computationally ex-pensive to solve.

The presence of discontinuities in the solutioncan lead to difficulties in solving the PDEs nu-merically, since derivatives are infinite at a pointof discontinuity, and finite difference approxima-tions to derivatives generally diverge. This haslead to the increased popularity of both finite vol-ume and Discontinuous Galerkin methods, whichare better able to robustly capture discontinuitiesin the solution. Methods designed to do this wellare often called shock capturing methods.

4.2 Inundation and the movingshoreline

Another computational challenge in modelingtsunamis, or any other geophysical flow overtopography, is the need to handle the movingboundary of the flow at the shoreline. Manyearly tsunami models did not capture this mov-ing boundary at all. Instead the equations weresolved over a fixed domain defined by the originalshoreline, and some boundary conditions imposedat this fixed boundary, such as an impermeablewall. While this cannot be used to model inunda-tion directly, it could still give some indication ofthe tsunami runup based on recording the depthand velocities along this wall boundary. Other

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mathematical or physical models were then usedto estimate inundation from these values.

Most tsunami models developed recently at-tempt to model inundation directly. For simpleproblems it may be possible to use a grid thatmoves with time so that one edge of the grid isalways along the shoreline. For realistic prob-lems this is generally infeasible since the shorelinecan be very complex and can break into piecesas islands or isolated pools of water form. Mosttsunami models instead use a fixed grid and im-plement some form of wetting and drying algo-rithm to keep track of which grid points or cellsare dry (h = 0) and which are wet (h > 0). Stan-dard approaches to approximating the PDEs typ-ically break down near the shoreline and a majorchallenge in developing tsunami models is to dealwith this case robustly and accurately, particu-larly since this is often the region of primary in-terest in terms of the model results.

4.3 Mesh refinement

Another challenge arises from the vast differencesin spatial scale between the ocean over whicha tsunmai propagates and a section of coastlinesuch as a harbor where the solution is of inter-est. In the shoreline region it may be necessaryto have a fine grid with perhaps 10 m or less be-tween grid points in order to resolve the flow ata scale that is useful. It is clearly impracticaland luckily also unnecessary to resolve the entireocean to this resolution. The wave length of atsunami is typically more than 100 km, so thegrid point spacing in the ocean can be more like1–10 km. Moreover we need even less resolutionover most of the ocean, particularly before thetsunami arrives.

To deal with the variation in spatial scales,virtually all tsunami codes use unequally spacedgrids, often by starting with a coarse grid overthe ocean and then refining portions of the gridto higher resolution where needed. Some modelsonly use static refinement, in which the grid doesnot change with time but has finer grids in re-gions of interest along the coast. Other computercodes use adaptive mesh refinement, in which theregions of refinement change with time, for exam-ple to follow the propagating tsunami with a finergrid near the wave peak than is used over the rest

of the ocean, and to refine near the coastal regionof interest only when the tsunami is approaching.

A related issue is the choice of time steps foradvancing the solution. Stability conditions gen-erally require that the time step multiplied by themaximum wave speed should be no greater thanthe width of a grid cell. This is because the ex-plicit methods that are typically used for solvinghyperbolic PDEs such as the shallow water equa-tions update the solution in each grid cell basedonly on data from the neighboring cells in eachtime step. If a wave can propagate more thanone grid cell in a timestep then the method be-comes unstable. This necessary condition for sta-bility is called the CFL condition after fundamen-tal work on the convergence of numerical methodsby Courant, Friedrichs, and Lewy in the 1920s.[Pointer to other entry?] For the shallow waterequations the wave speed is

√gh, which varies

dramatically from the shoreline where h ≈ 0 tothe deepest parts of the ocean, where h can reach10,000 m. Additional difficulties arise in imple-menting an adaptive mesh refinement algorithm:if the grid is refined in part of the domain by afactor of 10, say, in each spatial dimension, thentypically the time step must also be decreased bythe same factor. Hence for every time step on thecoarse grid it is necessary to take 10 time steps onthe finer grid, and information must be exchangedbetween the grids to maintain an accurate andstable solution near the grid interfaces.

4.4 Dispersive terms

In some situations, tsunamis are generated withshort wavelengths that are not sufficiently longrelative to the fluid depth for the shallow wa-ter equations to be valid. This most frequentlyhappens with smaller localized sources such as asubmarine landslide rather than large-scale earth-quakes. In this case it is often still possibleto use depth-averaged two-dimensional equations,but the equations obtained typically include ad-ditional terms involving higher order derivatives.These are generally dispersive terms that can bet-ter model the observed effect that waves with dif-ferent wavelength propagate at different speeds.

The introduction of higher order derivativestypically requires the use of implicit methods toefficiently solve the equations, since the stability

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constraint for an explicit method would generallyrequire a time step much smaller than desirable.Implicit methods result in coupled algebraic sys-tems of equations, often nonlinear, for the solu-tion at all of the grid points in each time step.

See [1] for a recent survey of tsunami sed-imentology, and [2] for a general introductionto probabilistic modeling of tsunamis. Somedetailed descriptions of numerical methods fortsunami simulation can be found, for example, in[3, 4, 5, 6, 7, 8]. The use of dispersive equationsfor modeling submarine landslides is discussed,for example, in [9].

Further Reading

1. J. Bourgeois. Geologic effects and records oftsunamis. In E. N. Bernard and A. R. Robin-son, editors, The Sea, volume 15, pages 55–92.Harvard University Press, 2009.

2. E. L. Geist, T. Parsons, U. S. ten Brink, andH. J. Lee. Tsunami probability. In E. N.Bernard and A. R. Robinson, editors, The Sea,volume 15, pages 201–235. Harvard UniversityPress, 2009.

3. F. X. Giraldo and T. Warburton. A high-ordertriangular discontinuous Galerkin oceanic shal-low water model. International Journal for Nu-merical Methods in Fluids, 56(7):899–925, 2008.

4. S. T. Grilli, M. Ioualalen, J. Asavanant, F. Shi,J. T. Kirby, and P Watts. Source constraintsand model simulation of the December 26, 2004,Indian Ocean Tsunami. Journal of Waterway,Port, Coastal, and Ocean Engineering, 133:414,2007.

5. Z. Kowalik, W. Knight, T. Logan, and P. Whit-more. Modeling of the global tsunami: Indone-sian Tsunami of 26 December 2004. Science ofTsunami Hazards, 23(1):40–56, 2005.

6. R. J. LeVeque, D. L. George, and M. J. Berger.Tsunami modeling with adaptively refined finitevolume methods. Acta Numerica, pages 211–289, 2011.

7. P.J. Lynett, T.R. Wu, and P.L.F. Liu. Model-ing wave runup with depth-integrated equations.Coastal Engineering, 46(2):89–107, 2002.

8. V. V. Titov and C. E. Synolakis. Numeri-cal modeling of tidal wave runup. J. Water-way, Port, Coastal, and Ocean Eng., 124:157–171, 1998.

9. P Watts, S Grilli, J Kirby, G. J. Fryer, and D. R.Tappin. Landslide tsunami case studies using aBoussinesq model and a fully nonlinear tsunamigeneration model. Nat. Haz. Earth Sys. Sci.,3:391–402, 2003.