Historically the finite volume methods have been developed for the numerical
integration of conservation laws. In this study we present some recent results
on the application of such schemes to dispersive PDEs. Namely, we solve
numerically a representative of Boussinesq type equations in view of important
applications to the coastal hydrodynamics. Numerical results of the runup of a
moderate wave onto a non-uniform beach are presented along with great lines of
the employed numerical method (see D. Dutykh et al. (2011) for more details).Comment: 8 pages, 6 figures, 18 references. This preprint is submitted to
  FVCA6 conference proceedings. Other author papers can be downloaded at
  http://www.lama.univ-savoie.fr/~dutykh

A Harten

AI Delis

AS Fokas

B Leer van

D Dutykh

DH Peregrine

E Audusse

JM Ghidaglia

M Simon

S Hibberd

S Tadepalli

T Benjamin

V.V. Titov

Y Kervella

English

arXiv

8 pages, 6 figures, 18 references. This preprint is submitted to FVCA6 conference proceedings. Other author papers can be downloaded at http://www.lama.univ-savoie.fr/~dutykh/International audienceHistorically the finite volume methods have been developed for the numerical integration of conservation laws. In this study we present some recent results on the application of such schemes to dispersive PDEs. Namely, we solve numerically a representative of Boussinesq type equations in view of important applications to the coastal hydrodynamics. Numerical results of the runup of a moderate wave onto a non-uniform beach are presented along with great lines of the employed numerical method (see D. Dutykh et al. (2011) for more details)

Dutykh, Denys

Katsaounis, Theodoros

Mitsotakis, Dimitrios

Hal - Université Grenoble Alpes

Dispersive wave runup on non-uniform shores

HAL Descartes

HAL Université de Savoie

Dispersive wave runup on non-uniform shoresDenys Dutykh, Theodoros Katsaounis, Dimitrios MitsotakisTo cite this version:Denys Dutykh, Theodoros Katsaounis, Dimitrios Mitsotakis. Dispersive wave runup onnon-uniform shores. Finite Volumes for Complex Applications VI Problems & Perspec-tives, 4, Springer Berlin Heidelberg, pp.389-397, 2011, 978-3-642-20671-9. <10.1007/978-3-642-20671-9 41>. <http://link.springer.com/chapter/10.1007/978-3-642-20671-9 41>. <hal-00553762v3>HAL Id: hal-00553762https://hal.archives-ouvertes.fr/hal-00553762v3Submitted on 5 Mar 2011HAL is a multi-disciplinary open accessarchive for the deposit and dissemination of sci-entific research documents, whether they are pub-lished or not. The documents may come fromteaching and research institutions in France orabroad, or from public or private research centers.L’archive ouverte pluridisciplinaire HAL, estdestine´e au de´poˆt et a` la diffusion de documentsscientifiques de niveau recherche, publie´s ou non,e´manant des e´tablissements d’enseignement et derecherche franc¸ais ou e´trangers, des laboratoirespublics ou prive´s.Distributed under a Creative Commons Attribution 4.0 International LicenseDispersive wave runup on non-uniform shoresDenys Dutykh and Theodoros Katsaounis and Dimitrios MitsotakisAbstract Historically the finite volume methods have been developed for the numer-ical integration of conservation laws. In this study we present some recent results onthe application of such schemes to dispersive PDEs. Namely, we solve numericallya representative of Boussinesq type equations in view of important applications tothe coastal hydrodynamics. Numerical results of the runup of a moderate wave ontoa non-uniform beach are presented along with great lines of the employed numericalmethod (see D. Dutykh et al. (2011) [6] for more details).Key words: dispersive wave, runup, Boussinesq equations, shallow waterMSC2010: 65M08, 76B151 IntroductionThe simulation of water waves in realistic and complex environments is a very chal-lenging problem. Most of the applications arise from the areas of coastal and navalengineering, but also from natural hazards assessment. These applications may re-quire the computation of the wave generation [5, 12], propagation [17], interactionwith solid bodies, the computation of long wave runup [16, 18] and even the ex-traction of the wave energy [15]. Issues like wave breaking, robustness of the nu-Denys DutykhLAMA, UMR 5127 CNRS, Universite´ de Savoie, Campus Scientifique, 73376 Le Bourget-du-LacCedex, France, e-mail: Denys.Dutykh@univ-savoie.frTheodoros KatsaounisDepartment of Applied Mathematics, University of Crete, Heraklion, 71409 Greece Inst. of App.and Comp. Math. (IACM), FORTH, Heraklion, 71110, Greece e-mail: thodoros@tem.uoc.grDimitrios MitsotakisIMA, University of Minnesota, Minneapolis MN 55455, USA e-mail: dmitsot@gmail.com12 D. Dutykh, Th. Katsaounis and D. Mitsotakismerical algorithm in wet-dry processes along with the validity of the mathematicalmodels in the near-shore zone are some basic problems in this direction [11]. Dur-ing past several decades the classical Nonlinear Shallow Water Equations (NSWE)have been essentially employed to face these problems [7]. Mathematically, theseequations represent a system of conservation laws describing the propagation of in-finitely long waves with a hydrostatic pressure assumption. The wave breaking phe-nomenon is commonly assimilated to the formation of shock waves (or hydraulicjumps) which is a common feature of hyperbolic PDEs. Consequently, the finitevolume (FV) method has become the method of choice for these problems due to itsexcellent intrinsic conservative and shock-capturing properties [3, 7].In the present article we report on recent results concerning the extension of thefinite volume method to dispersive wave equations steming essentially from waterwave modeling [14, 4, 6].2 Mathematical model and numerical methodsConsider a cartesian coordinate system in two space dimensions (x,z) to simplifynotations. The z-axis is taken vertically upwards and the x-axis is horizontal andcoincides traditionally with the still water level. The fluid domain is bounded belowby the bottom z = −h(x) and above by the free surface z = η(x, t). Below we willalso need the total water depth H(x, t) := h(x)+η(x, t). The flow is supposed tobe incompressible and the fluid is inviscid. An additional assumption of the flowirrotationality is made as well.In the pioneering work of D.H. Peregrine (1967) [14] the following system ofBoussinesq type equations has been derived:ηt +((h+η)u)x= 0, (1)ut + uux+ gηx− h2(hu)xxt +h26 uxxt = 0, (2)where u(x, t) is the depth averaged fluid velocity, g is the gravity acceleration andunderscripts (ux, ηt ) denote partial derivatives.In our recent study [6] we proposed an improved version of this system whichcontains higher order nonlinear terms which should be neglected from asymptoticpoint of view and can be written in conservative variables (H,Q) = (H,Hu) as:Ht +Qx = 0, (3)((1+ 13H2x −16 HHxx)Qt − 13H2Qxxt − 13 HHxQxt)+(Q2H+g2H2)x= gHhx. (4)Dispersive wave runup on non-uniform shores 3Obviously the linear characteristics of both systems (1), (2) and (3), (4) coincidesince they differ only by nonlinear terms.However, this modification has several important implications onto structuralproperties of the obtained system. First of all, the magnitude of the dispersive termstends to zero when we approach the shoreline H → 0. This property corresponds toour physical representation of the wave shoaling and runup process. On the otherhand, the resulting system becomes invariant under vertical translations (subgroupG5 in Theorem 4.2, T. Benjamin & P. Olver (1982) [2]):z← z+ d, η ← η− d, h← h+ d, u← u, (5)where d is some constant. This property is straightforward to check since we useonly the total water depth variable H = h+η which remains invariant under trans-formation (5).Remark 1 In this paper we will consider the initial-boundary value problem posedin a bounded domain I = [b1,b2] with reflective boundary conditions. In this caseone needs to impose boundary conditions only in one of the two dependent variables,cf. [8]. In the case of reflective boundary conditions it is sufficient to take u(b1, t) =u(b2, t) = 0.2.1 Finite volume discretizationLet T = {xi}, i ∈ Z denotes a partition of R into cells Ci = (xi− 12 ,xi+ 12 ) wherexi = (xi+ 12+xi− 12 )/2 denotes the midpoint of Ci. Let ∆xi = xi+ 12 −xi− 12 be the lengthof the cell Ci, ∆xi+ 12 = xi+1− xi. (Here, we consider only uniform grids with ∆xi =∆xi+ 12 = ∆x.)The governing equations (3), (4) can be recast in the following vector form:[D(vt)]+ [F(v)]x = S(v),whereD(vt) =(Ht(1+ 13 H2x − 16 HHxx)Qt − 13 H2Qxxt − 13 HHxQxt), (6)F(v) =(QQ2H +g2 H2), S(v) =(0gHhx). (7)We denote by Hi and Ui the corresponding cell averages. To discretize the dis-persive terms in (6) we consider the following approximations:4 D. Dutykh, Th. Katsaounis and D. Mitsotakis1∆x∫ xi+ 12xi− 12[1+13 (Hx)2− 16HHxx]Q dx≈(1+ 13(Hi+1−Hi−12∆x)2− 16HiHi+1− 2Hi+Hi−1∆x2)Qi,1∆x∫ xi+ 12xi− 1213 HHxQx dx≈13 HiHi+1−Hi−12∆xQi+1−Qi−12∆x ,1∆x∫ xi+ 12xi− 1213 H2Qxx dx≈ 13 H2iQi+1− 2Qi+Qi−1∆x2 .We note that we approximate the reflective boundary conditions by taking thecell averages of u on the first and the last cell to be u0 = uN+1 = 0. We do notimpose explicitly boundary conditions on H. The reconstructed values on the firstand the last cell are computed using neighboring ghost cells and taking odd and evenextrapolation for u and H respectively. These specific boundary conditions appearedto reflect incident waves on the boundaries while conserving the mass.This discretization leads to a linear system with tridiagonal matrix denoted by Lthat can be inverted efficiently by a variation of Gauss elimination for tridiagonalsystems with computational complexity O(n), n-being the dimension of the system.We note that on the dry cells the matrix becomes diagonal since Hi is zero on drycells. For the time integration the explicit third-order TVD-RK method is used. Inthe numerical experiments we observed that the fully discrete scheme is stable andpreserves the positivity of H during the runup under a mild restriction on the timestep ∆ t.Therefore, the semidiscrete problem of (6) - (7) is written as a system of ODEsin the form:Livit +1∆x (Fi+ 12 −Fi− 12 ) =1∆xSi,where Li is the i−th row of matrix L and Fi+ 12 can be chosen as one of the numer-ical flux functions [6] (in computations presented below we choose the FVCF flux[9]). In the sequel we will use the KT and the CF numerical fluxes. In this case theJacobian of F is given by the matrixA =(0 1gH− (Q/H)2 2Q/H),and the eigenvalues are λ1,2 = Q/H±√gH. Therefore, the characteristic numericalflux [9] takes the formFi+ 12=F(VLi+ 12)+F(VRi+ 12)2−U(µ)F(VRi+ 12)−F(VLi+ 12 )2,where µ = (µ1,µ2)T are the Roe average values,Dispersive wave runup on non-uniform shores 5µ1 =HLi+ 12+HRi+ 122, µ2 =√HLi+ 12ULi+ 12+√HRi+ 12URi+ 12√HLi+ 12+√HRi+ 12andU(µ) =(s2(µ2+c)−s1(µ2−c)2cs1−s22c(s2−s1)(µ22−c2)2cs1(µ2+c)−s2(µ2−c)2c), c =√gµ1, si = sign(λi).For more details on the discretization and reconstruction procedures, (that arebased on the hydrostatic reconstraction, [1]), we refer to our complete work on thissubject [6].3 Numerical resultsIn the present section we show a numerical simulation of a solitary wave runup ontoa non-uniform sloping beach. More precisely, we add a small pond along the slope.As our results indicate, this small complication is already sufficient to develop someinstabilities which remain controlled in our simulations.As an initial condition we used an approximate solitary wave solution of thefollowing form:η0(x) = Assech2(λ (x− x0)), u0(x) =−cs η0(x)1+η0(x) ,where As is the amplitude relative to the constant water depth taken to be unity inour study. The solitary wave speed cs along with the wavelength λ are given here:λ =√3As4(1+As), , cs =√g√6(1+As)√3+ 2As·√(1+As) log(1+As)−AsAs.The solitary wave is centered initially at x0 = 10.62 and has amplitude As = 0.08.The constant slope β is equal to 2.88◦. The sketch of the computational domain canbe found in [6].In numerical simulations presented below we used a uniform space discretizationwith ∆x = 0.025 and very fine time step ∆ t = ∆x/100 to guarantee the accuracy andstability during the whole simulation.Snapshots of numerical results are presented on Figures 1 – 6. We present simul-taneously three different computational results:• Modified Peregrine system solved with UNO2 reconstruction [10]• The same system with classical MUSCL TVD2 scheme [13]• Nonlinear Shallow Water Equations (NSWE) with UNO2 scheme [10]6 D. Dutykh, Th. Katsaounis and D. Mitsotakis−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(a) t = 1 s−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(b) t = 3 sFig. 1 Solitary wave aproaching a sloping beach with a pond.−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(a) t = 3.5 s−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(b) t = 4 sFig. 2 Beginning of the pond inundation.−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(a) t = 5 s−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(b) t = 5.5 sFig. 3 A part of the wave mass is trapped in the pond volume.Dispersive wave runup on non-uniform shores 7−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(a) t = 6 s−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(b) t = 6.5 sFig. 4 Wave oscillations in the pond.−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(a) t = 7 s−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)(b) t = 8 sFig. 5 Stabilization of wave oscillations.−2 0 2 4 6 8 10−0.2−0.100.10.20.3xη  Boussinesq (UNO)Boussinesq (TVD)Shallow water (UNO)Fig. 6 The whole system is tending to the rest position (t = 10 s).Surprisingly good agreement was obtained among all three numerical models.Presumably, the complex runup process under consideration is governed essen-tially by nonlinearity. However, on Figures 1(b) and 2(a) the amplitude predictedby NSWE is slightly overestimated.On Figures 3(b) – 4(b) some oscillations (due to the small-dispersion effect char-acterizing dispersive wave breaking procedures) can be observed. However, theiramplitude remains small for all times and does not produce any blow up phenom-8 D. Dutykh, Th. Katsaounis and D. Mitsotakisena. Later these oscillations decay tending gradually to the “lake at the rest” state(see Figures 5, 6).In the specific experiment a friction term could be beneficial to reduce the ampli-tude of oscillations (or damp them out completely). However, we prefer to presentthe computational results of our model without adding any ad-hoc term to show itsoriginal performance.4 ConclusionsIn this study we presented an improved version of the Peregrine system which is par-ticularly suited for the simulation of dispersive waves runup. This system allows forthe description of higher amplitude waves due to improved nonlinear characteristics.Better numerical stability properties have been obtained since most of the disper-sive terms tend to zero when we approach the shoreline. Consequently, our modelnaturally degenerates to classical Nonlinear Shallow Water Equations (NSWE) forwhich the runup simulation technology is completely mastered nowadays. Howeverwe underline that there is no artificial parameter to turn off dispersive terms. Theirimportance is naturally governed by the underlying physical process.Acknowledgements D. Dutykh acknowledges the support from French Agence Nationale dela Recherche, project MathOcean (Grant ANR-08-BLAN-0301-01) and Ulysses Program of theFrench Ministry of Foreign Affairs under the project 23725ZA. The work of Th. Katsaounis waspartially supported by European Union FP7 program Capacities(Regpot 2009-1), through ACMAC(http://acmac.tem.uoc.gr).References1. Audusse, E., Bouchut, F., Bristeau, O., Klein, R., Perthame, B.: A fast and stable well-balancedscheme with hydrostatic reconstruction for shallow water flows. SIAM J. of Sc. Comp. 25,2050–2065 (2004)2. Benjamin, T., Olver, P.: Hamiltonian structure, symmetries and conservation laws for waterwaves. J. Fluid Mech 125, 137–185 (1982)3. Delis, A.I., Katsaounis, T.: Relaxation schemes for the shallow water equations. Int. J. Numer.Meth. Fluids 41, 695–719 (2003)4. Dutykh, D., Dias, F.: Dissipative Boussinesq equations. C. R. Mecanique 335, 559–583 (2007)5. Dutykh, D., Dias, F.: Water waves generated by a moving bottom. In: A. Kundu (ed.) Tsunamiand Nonlinear waves. Springer Verlag (Geo Sc.) (2007)6. Dutykh, D., Katsaounis, T., Mitsotakis, D.: Finite volume schemes for dispersive wavepropagation and runup. Accepted to Journal of Computational Physics http://hal.archives-ouvertes.fr/hal-00472431/ (2011)7. Dutykh, D., Poncet, R., Dias, F.: Complete numerical modelling of tsunami waves: generation,propagation and inundation. Submitted http://arxiv.org/abs/1002.4553 (2010)8. Fokas, A.S., Pelloni, B.: Boundary value problems for Boussinesq type systems. Math. Phys.Anal. Geom. 8, 59–96 (2005)Dispersive wave runup on non-uniform shores 99. Ghidaglia, J.M., Kumbaro, A., Coq, G.L.: On the numerical solution to two fluid models viacell centered finite volume method. Eur. J. Mech. B/Fluids 20, 841–867 (2001)10. Harten, A., Osher, S.: Uniformly high-order accurate nonscillatory schemes, I. SIAM J. Nu-mer. Anal. 24, 279–309 (1987)11. Hibberd, S., Peregrine, D.: Surf and run-up on a beach: a uniform bore. J. Fluid Mech. 95,323–345 (1979)12. Kervella, Y., Dutykh, D., Dias, F.: Comparison between three-dimensional linear and nonlin-ear tsunami generation models. Theor. Comput. Fluid Dyn. 21, 245–269 (2007)13. van Leer, B.: Towards the ultimate conservative difference scheme V: a second order sequelto Godunov’ method. J. Comput. Phys. 32, 101–136 (1979)14. Peregrine, D.H.: Long waves on a beach. J. Fluid Mech. 27, 815–827 (1967)15. Simon, M.: Wave-energy extraction by a submerged cylindrical resonant duct. Journal of FluidMechanics 104, 159–187 (1981)16. Tadepalli, S., Synolakis, C.E.: The run-up of N-waves on sloping beaches. Proc. R. Soc. Lond.A 445, 99–112 (1994)17. Titov, V., Gonza´lez, F.: Implementation and testing of the method of splitting tsunami (MOST)model. Tech. Rep. ERL PMEL-112, Pacific Marine Environmental Laboratory, NOAA (1997)18. Titov, V.V., Synolakis, C.E.: Numerical modeling of tidal wave runup. J. Waterway, Port,Coastal, and Ocean Engineering 124, 157–171 (1998)The paper is in final form and no similar paper has been or is being submitted elsewhere.

Denys Dutykh

Theodoros Katsaounis

Dimitrios Mitsotakis

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Dispersive wave runup on non-uniform shores
Denys Dutykh and Theodoros Katsaounis and Dimitrios Mitsotakis
Original Citation:
Dutykh, Denys and Katsaounis, Theodoros and Mitsotakis, Dimitrios
(2011)
Dispersive wave runup on non-uniform shores.
Finite volumes for complex application VI Problems & Perspectives, 4. pp. 389-397. ISSN 2190-
5614
This version is available at: http://preprints.acmac.uoc.gr/32/
Available in ACMAC’s PrePrint Repository: February 2012
ACMAC’s PrePrint Repository aim is to enable open access to the scholarly output of ACMAC.
http://preprints.acmac.uoc.gr/Dispersive wave runup on non-uniform shores
Denys Dutykh and Theodoros Katsaounis and Dimitrios Mitsotakis
AbstractHistoricallytheﬁnitevolumemethodshavebeendevelopedforthenumer-
ical integration of conservationlaws. In this study we present some recent results on
the application of such schemes to dispersive PDEs. Namely, we solve numerically
ar e p r e s e n t a t i v eo fB o u s s i n e s qt y p ee q u a t i o n si nv i e wo fi m p ortant applications to
the coastal hydrodynamics.Numerical results of the runup ofam o d e r a t ew a v eo n t o
an o n - u n i f o r mb e a c ha r ep r e s e n t e da l o n gw i t hg r e a tl i n e so ft he employednumerical
method (see D. Dutykh et al. (2011) [7] for more details).
1I n t r o d u c t i o n
The simulation of water waves in realistic and complexenvironmentsis a very chal-
lenging problem. Most of the applications arise from the areas of coastal and naval
engineering, but also from natural hazards assessment. These applications may re-
quire the computation of the wave generation [6, 13], propagation [18], interaction
with solid bodies, the computation of long wave runup [17, 19]a n de v e nt h ee x t r a c -
tion of the wave energy [16]. Issues like wave breaking,robustness of the numerical
algorithminwet-dryprocessesalongwiththevalidityofthemathematicalmodelsin
the near-shore zone are some basic problems in this direction[ 1 2 ] .D u r i n gp a s ts e v -
eral decades the classical Nonlinear Shallow Water Equations (NSWE) have been
essentially employed to face these problems [1, 8]. Mathematically, these equations
Denys Dutykh
LAMA, UMR 5127 CNRS, Universit´ e de Savoie, Campus Scientiﬁque, 73376 Le Bourget-du-Lac
Cedex, France, e-mail: Denys.Dutykh@univ-savoie.fr
Theodoros Katsaounis
Department of Applied Mathematics, University of Crete, Heraklion, 71409 Greece Inst. of App.
and Comp. Math. (IACM), FORTH, Heraklion, 71110, Greece e-mail: thodoros@tem.uoc.gr
Dimitrios Mitsotakis
IMA, University of Minnesota, Minneapolis MN 55455, USA e-mail: dmitsot@gmail.com
12 D. Dutykh, Th. Katsaounis and D. Mitsotakis
representa system ofconservationlawsdescribingthepropagationofinﬁnitelylong
waves with a hydrostatic pressure assumption. The wave breaking phenomenon is
commonly assimilated to the formation of shock waves (or hydraulic jumps) which
is a common feature of hyperbolic PDEs. Consequently, the ﬁnite volume (FV)
method has become the method of choice for these problems due to its excellent
intrinsic conservative and shock-capturing properties [1,4 ,8 ] .
In the present article we report on recent results concerningt h ee x t e n s i o no ft h e
ﬁnite volume method to dispersive wave equations steming essentially from water
wave modeling [15, 5, 7].
2M a t h e m a t i c a lm o d e la n dn u m e r i c a lm e t h o d s
Consider a cartesian coordinate system in two space dimensions (x,z) to simplify
notations. The z-axis is taken vertically upwards and the x-axis is horizontal and
coincides traditionally with the still water level. The ﬂuidd o m a i ni sb o u n d e db e l o w
by the bottom z =  h(x) and above by the free surface z = η(x,t).B e l o ww ew i l l
also need the total water depth H(x,t) := h(x)+η(x,t).T h eﬂ o wi ss u p p o s e dt o
be incompressible and the ﬂuid is inviscid. An additional assumption of the ﬂow
irrotationality is made as well.
In the pioneering work of D.H. Peregrine (1967) [15] the following system of
Boussinesq type equations has been derived:
ηt +
 
(h+η)u
 
x = 0, (1)
ut +uux+gηx 
h
2
(hu)xxt +
h2
6
uxxt = 0, (2)
where u(x,t) is the depth averaged ﬂuid velocity, g is the gravity acceleration and
underscripts (ux, ηt)d e n o t ep a r t i a ld e r i v a t i v e s .
In our recent study [7] we proposed an improved version of thiss y s t e mw h i c h
contains higher order nonlinear terms which should be neglected from asymptotic
point of view and can be written in conservative variables (H,Q)=( H,Hu) as:
Ht +Qx = 0, (3)
  
1+
1
3
H2
x  
1
6
HHxx
 
Qt  
1
3
H2Qxxt  
1
3
HHxQxt
 
+
 Q2
H
+
g
2
H2
 
x
= gHhx. (4)
Obviously the linear characteristics of both systems (1), (2) and (3), (4) coincide
since they differ only by nonlinear terms.
However, this modiﬁcation has several important implications onto structural
properties of the obtained system. First of all, the magnitude of the dispersive terms
tends to zero when we approach the shoreline H   0. This property corresponds to
our physical representation of the wave shoaling and runup process. On the other
hand, the resulting system becomes invariant under verticalt r a n s l a t i o n s( s u b g r o u pDispersive wave runup on non-uniform shores 3
G5 in Theorem 4.2, T. Benjamin & P. Olver (1982) [3]):
z   z+d, η   η  d, h   h+d, u   u, (5)
where d is some constant. This property is straightforward to check since we use
only the total water depth variable H = h+η which remains invariant under trans-
formation (5).
Remark 1 In this paper we will consider the initial-boundaryvalue problem posed
in a bounded domain I =[ b1,b2] with reﬂective boundary conditions. In this case
oneneedstoimposeboundaryconditionsonlyinoneofthetwodependentvariables,
cf. [9]. In the case of reﬂective boundary conditions it is sufﬁcient to take u(b1,t)=
u(b2,t)=0.
2.1 Finite volume discretization
Let T = {xi}, i   Z denotes a partition of R into cells Ci =( xi  1
2
,xi+ 1
2
) where
xi =(xi+ 1
2
+xi  1
2
)/2denot est hemi dpoi ntofCi.L e tΔxi =xi+ 1
2
 xi  1
2
be thelength
of the cellCi, Δxi+ 1
2
= xi+1 xi.( H e r e ,w ec o n s i d e ro n l yu n i f o r mg r i d sw i t hΔxi =
Δxi+ 1
2
= Δx.)
The governingequations (3), (4) can be recast in the following vector form:
[D(vt)]+[F(v)]x = S(v),
where
D(vt)=
 
Ht
(1+ 1
3H2
x   1
6HHxx)Qt   1
3H2Qxxt   1
3HHxQxt
 
, (6)
F(v)=
 
Q
Q2
H +
g
2H2
 
, S(v)=
 
0
gHhx
 
. (7)
We denote by Hi and Ui the corresponding cell averages. To discretize the dis-
persive terms in (6) we consider the following approximations:
1
Δx
  x
i+ 1
2
x
i  1
2
 
1+
1
3
(Hx)2 
1
6
HHxx
 
Qdx 
 
1+
1
3
 
Hi+1 Hi 1
2Δx
 2
 
1
6
Hi
Hi+1 2Hi+Hi 1
Δx2
 
Qi,
1
Δx
  x
i+ 1
2
x
i  1
2
1
3
HHxQx dx 
1
3
Hi
Hi+1 Hi 1
2Δx
Qi+1 Qi 1
2Δx
,4 D. Dutykh, Th. Katsaounis and D. Mitsotakis
1
Δx
  x
i+ 1
2
x
i  1
2
1
3
H2Qxx dx 
1
3
H2
i
Qi+1 2Qi+Qi 1
Δx2 .
We note that we approximate the reﬂective boundary conditions by taking the
cell averages of u on the ﬁrst and the last cell to be u0 = uN+1 = 0. We don’timpose
explicitly boundary conditions on H.T h er e c o n s t r u c t e dv a l u e so nt h eﬁ r s ta n dt h e
last cell are computed using neighboring ghost cells and taking odd and even ex-
trapolation for u and H respectively. These speciﬁc boundary conditions appeared
to reﬂect incident waves on the boundaries while conserving the mass.
This discretization leads to a linear system with tridiagonal matrix denoted by L
that can be inverted efﬁciently by a variation of Gauss elimination for tridiagonal
systems with computationalcomplexity O(n), n-being the dimension of the system.
We note that on the dry cells the matrix becomes diagonal since Hi is zero on dry
cells. For the time integration the explicit third-order TVD-RK method is used. In
the numerical experiments we observed that the fully discrete scheme is stable and
preserves the positivity of H during the runup under a mild restriction on the time
step Δt.
Therefore, the semidiscrete problem of (6) - (7) is written asas y s t e mo fO D E s
in the form:
Livit +
1
Δx
(Fi+ 1
2
 Fi  1
2
)=
1
Δx
Si,
where Li is the i th row of matrix L and Fi+ 1
2
can be chosen as one of the numer-
ical ﬂux functions [7] (in computations presented below we choose the FVCF ﬂux
[10]). In the sequel we will use the KT and the CF numerical ﬂuxes. In this case the
Jacobian of F is given by the matrix
A =
 
01
gH (Q/H)2 2Q/H
 
,
and the eigenvaluesare λ1,2 =Q/H±
 
gH.T h e r e f o r e ,t h ec h a r a c t e r i s t i cn u m e r i c a l
ﬂux [10] takes the form
Fi+ 1
2
=
F(VL
i+ 1
2
)+F(VR
i+ 1
2
)
2
 U(µ)
F(VR
i+ 1
2
) F(VL
i+ 1
2
)
2
,
where µ =( µ1,µ2)T are the Roe average values,
µ1 =
HL
i+ 1
2
+HR
i+ 1
2
2
, µ2 =
 
HL
i+ 1
2
UL
i+ 1
2
+
 
HR
i+ 1
2
UR
i+ 1
2  
HL
i+ 1
2
+
 
HR
i+ 1
2
and
U(µ)=
 
s2(µ2+c) s1(µ2 c)
2c
s1 s2
2c
(s2 s1)(µ2
2 c2)
2c
s1(µ2+c) s2(µ2 c)
2c
 
, c =
 
gµ1, si = sign(λi).Dispersive wave runup on non-uniform shores 5
For more details on the discretization and reconstruction procedures, (that are
based on the hydrostatic reconstraction, [2]), we refer to our complete work on this
subject [7].
3N u m e r i c a lr e s u l t s
In the present section we show a numerical simulation of a solitary wave runup onto
an o n - u n i f o r ms l o p i n gb e a c h .M o r ep r e c i s e l y ,w ea d das m a l lp ond along the slope.
As our results indicate, this small complicationis already sufﬁcient to develop some
instabilities which remain controlled in our simulations.
As an initial condition we used an approximate solitary wave solution of the
following form:
η0(x)=Assech2 
λ(x x0)
 
, u0(x)= cs
η0(x)
1+η0(x)
,
where As is the amplitude relative to the constant water depth taken tob e1i no u r
study. The solitary wave speed cs along with the wavelength λ are given here:
λ =
 
3As
4(1+As)
,, cs =
 
g
 
6(1+As)
 
3+2As
·
 
(1+As)log(1+As) As
As
.
The solitary wave is centered initially at x0 = 10.62 and has amplitude As = 0.08.
The constant slope β is equal to 2.88 .T h es k e t c ho ft h ec o m p u t a t i o n a ld o m a i nc a n
be found in [7].
Innumericalsimulationspresentedbelow we used a uniformspace discretization
with Δx=0.025and veryﬁne time step Δt =Δx/100to guaranteethe accuracyand
stability during the whole simulation.
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(a) t = 1s
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(b) t = 3s
Fig. 1 Solitary wave aproaching a sloping beach with a pond.6 D. Dutykh, Th. Katsaounis and D. Mitsotakis
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(a) t = 3.5s
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(b) t = 4s
Fig. 2 Beginning of the pond inundation.
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(a) t = 5s
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(b) t = 5.5s
Fig. 3 Ap a r to ft h ew a v em a s si st r a p p e di nt h ep o n dv o l u m e .
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(a) t = 6s
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(b) t = 6.5s
Fig. 4 Wave oscillations in the pond.
Snapshots of numerical results are presented on Figures 1 – 6.W ep r e s e n ts i m u l -
taneously three different computational results:
• Modiﬁed Peregrine system solved with UNO2 reconstruction [11]
• The same system with classical MUSCL TVD2 scheme [14]
• Nonlinear Shallow Water Equations (NSWE) with UNO2 scheme [11]
Surprisingly good agreement was obtained among all three numerical models.
Presumably, the complex runup process under consideration is governed essen-Dispersive wave runup on non-uniform shores 7
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(a) t = 7s
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
(b) t = 8s
Fig. 5 Stabilization of wave oscillations.
−2 0 2 4 6 8 10
−0.2
−0.1
0
0.1
0.2
0.3
x
 
 
 
Boussinesq (UNO)
Boussinesq (TVD)
Shallow water (UNO)
Fig. 6 The whole system is tending to the rest position (t = 10 s).
tially by nonlinearity. However, on Figures 1(b) and 2(a) thea m p l i t u d ep r e d i c t e d
by NSWE is slightly overestimated.
On Figures 3(b) – 4(b) some oscillations (due to the small-dispersion effect char-
acterizing dispersive wave breaking procedures) can be observed. However, their
amplitude remains small for all times and does not produce anyb l o wu pp h e n o m -
ena. Later these oscillations decay tending gradually to the“ l a k ea tt h er e s t ”s t a t e
(see Figures 5, 6).
In the speciﬁc experimenta friction term could be beneﬁcialto reduce the ampli-
tude of oscillations (or damp them out completely). However,w ep r e f e rt op r e s e n t
the computational results of our model without adding any ad-hoc term to show its
original performance.
4C o n c l u s i o n s
Inthisstudywe presentedanimprovedversionofthePeregrinesystemwhichispar-
ticularly suited for the simulation of dispersive waves runup. This system allows for
thedescriptionofhigheramplitudewaves dueto improvednonlinearcharacteristics.
Better numerical stability properties have been obtained since most of the disper-8 D. Dutykh, Th. Katsaounis and D. Mitsotakis
sive terms tend to zero when we approach the shoreline. Consequently, our model
naturally degenerates to classical Nonlinear Shallow WaterE q u a t i o n s( N S W E )f o r
which the runup simulation technologyis completely mastered nowadays. However
we underline that there is no artiﬁcial parameter to turn off dispersive terms. Their
importance is naturally governed by the underlying physicalp r o c e s s .
Moreover we presented some numerical results on the wave runup onto a com-
plex beach containing a pond. Even in this stiff case the numerical model at hand
produced stable and physical results, thus validating modiﬁcation of the Peregrine
system.
Acknowledgements D. Dutykh acknowledges the support from French Agence Nationale de
la Recherche, project MathOcean (Grant ANR-08-BLAN-0301-01) and Ulysses Program of the
French Ministry of Foreign Affairs under the project 23725ZA. The work of Th. Katsaounis was
partially supported by European Union FP7 program Capacities(Regpot 2009-1), through ACMAC
(http://acmac.tem.uoc.gr).
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Dispersive wave runup on non-uniform shores

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