ACMAC

Helzel, Christian

Otto, Felix

Tzavaras, A. E.

English

A kinetic model for the sedimentation of rod-like particles

Dutykh, Denys

Katsaounis, Theodoros

Mitsotakis, Dimitrios

ACMAC’s PrePrint Repository
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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