A numerical model preserving nontrivial steady-state solutions for predicting waves run-up on coastal areas Hasan Karjouna Abdelaziz Beljadidab

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A numerical model preserving nontrivial steady-state solutions

for predicting waves run-up on coastal areas

Hasan Karjouna, Abdelaziz Beljadida,b,∗

aMohammed VI Polytechnic University, Green City, Morocco

bUniversity of Ottawa, Ottawa, Canada

Abstract

In this study, a numerical model preserving a class of nontrivial steady-state solutions is pro-

posed to predict waves propagation and waves run-up on coastal zones. The numerical model

is based on the Saint-Venant system with source terms due to variable bottom topography

and bed friction eﬀects. The resulting nonlinear system is solved using a Godunov-type ﬁnite

volume method on unstructured triangular grids. A special piecewise linear reconstruction

of the solution is implemented with a correction technique to ensure the accuracy of the

method and the positivity of the computed water depth. Eﬃcient semi-implicit techniques

for the friction terms and a well-balanced formulation for the bottom topography are used

to exactly preserve stationary steady-state s solutions. Moreover, we prove that the numer-

ical scheme preserves a class of nontrivial steady-state solutions. To validate the proposed

numerical model against experiments, we ﬁrst demonstrate its ability to preserve nontrivial

steady-state solutions and then we model several laboratory experiments for the predic-

tion of waves run-up on sloping beaches. The numerical simulations are in good agreement

with laboratory experiments which conﬁrms the robustness and accuracy of the proposed

numerical model in predicting waves propagation on coastal areas.

Keywords: Coastal areas; wave run-up/run-down; shallow water model; ﬁnite volume

method; well-balanced discretization, positivity preserving property.

1. Introduction

Coastal areas involve several complex natural processes such as surface ﬂows, sediment

transport, soil erosion and moving shorelines. The propagation of waves on coastal areas

near urban zones can have negative environmental impacts and cause considerable damages

[1,2,3]. Understanding the dynamics of ﬂow waves and predict its eﬀects on coastal areas

is necessary for developing solutions for sustainable water management and reducing water-

related hazard including environment risks. Coastal wave propagation is mainly aﬀected

by the complex geometry of nearshore zones and the bottom topography. The rugged

topography can cause many wave transformations such as wave refraction, diﬀraction, and

breaking as the wave approaches the shoreline [4,5,6]. Wave run-up at the coastline

is mainly depends on the oﬀshore wave conditions such as height, length, and velocity

Preprint submitted October 5, 2022

arXiv:2210.01499v1 [math.NA] 4 Oct 2022

of waves and on the coastal geometry and topography [7]. Several previous studies have

been devoted to predict waves propagation and determine the height of run-up along the

coastlines [8,9,10,11,12,13,14,15,16,17,18]. Synolakis [19,20] performed theoretical

and experimental study to analyze the evolution of non-breaking and breaking solitary waves

and approximated the wave maximum run-up, as well as the breaking criterion when the

wave climbs up the sloping breach. Madsen and Mei [21] investigated the transformation of

the solitary wave over an uneven bottom topography and showed that the wave height rises

depending on the slope and its initial height. Kaplan [22] studied the evolution of periodic

waves and derived an empirical formula for wave maximum run-up over a sloping beach.

The non-linear shallow water equations are among the commonly used models to describe

the propagation of waves over variable bottom topography. These equations are derived from

a depth-averaged integration of the 3D incompressible Navier–Stokes equations under the

hydrostatic pressure assumption [23]. They are widely used for modeling water ﬂows in

lakes, rivers, and coastal areas [24,25,26,27,28,29,30]. Titov and Synolakis [31] tested

the eﬃciency of the shallow water equations for modeling the evolution of breaking and non-

breaking solitary waves on sloping beaches based on data from laboratory experiments. Delis

et al. [13] developed a numerical model based on shallow water equations and investigated

its accuracy to simulate long waves. Hu et al. [32] used nonlinear shallow water equations

to perform numerical simulations of wave overtopping of coastal structures. Brocchini and

Dodd [33] modeled nearshore ﬂows using nonlinear shallow water equations and analyzed

the interdependence between physical phenomena, model equations and numerical methods.

Various numerical methods have been developed for solving the shallow water equations

modeling ﬂows over variable bottom topography [34,35,36,37,38,39]. The ﬁnite volume

technique is one of the most popular tools used for solving the system of shallow water

equations due to its capability to conserve mass and momentum, and accurately computes

solutions with sharp gradients. Central-upwind ﬁnite volume methods are among the most

eﬃcient numerical tools developed for solving the system of conservation laws [40,41]. The

central-upwind schemes are Godunov-type free Riemann solver, which are based on the

information obtained from the local speeds of wave propagation to approximate the nu-

merical ﬂuxes at the cell interfaces. Many extensions of these schemes have been proposed

for the system of shallow water ﬂow due to their robustness, high resolution and simplicity

[42,43,44,45,25,46,47]. Furthermore, central-upwind schemes can be well-balanced and

positivity preserving, that is, they accurately compute stationary steady solutions of the

model and maintain the non-negativity of the computed water depth at the discrete level

[48,49,50].

In this study, the depth-averaged 2-D shallow water equations are used for modeling

waves propagation and waves run-up at the coastlines. The source terms due to variable

bottom topography and bed friction eﬀect are taken into account. The shallow water system

is solved using an unstructured ﬁnite volume method based on central-upwind techniques

[50,41]. We used eﬃcient semi-implicit techniques for the friction term and a well-balanced

discretization for the source term due to variable topography. Beside the well-balanced

property of the lack at rest, we established that the numerical scheme preserves nontrivial

steady-state solutions over a slanted surface. Furthermore, a piecewise linear approximation

of the solution is implemented at the discrete level to ensure the accuracy of numerical

scheme while guaranteeing the non-negativity of the computed water depth. Numerical

simulations are performed to test the eﬃciency and the capability of the numerical model

for predicting waves propagation and waves run-up on coastal areas.

The outline of this paper is as follows. We present the shallow water model with source

terms due to variable bottom topography and bed friction eﬀects in Section 2. The result-

ing nonlinear system of the shallow water ﬂow model is solved using unstructured ﬁnite

volume central-upwind technique on triangular meshes in Section 3. In Section 4, we per-

form numerical simulations to validate the proposed numerical model and test its capability

for simulating waves propagation on coastlines. The numerical model is validated against

laboratory experimental data and we perform numerical simulations to predict waves prop-

agation along a sloping beach with complex bottom topography. Finally, some concluding

remarks are provided in Section 5.

2. Governing equations

2.1. Shallow water equations

In this study, we consider the shallow water equations for modeling waves propagation

and waves run-up on the coastal areas. Let h(x, y, t)[m]be the water depth above the bottom

topography B(x, y)[m], and u= (u(x, y, t), v(x, y, t))T[m/s]be the velocity ﬁeld of the ﬂow

as shown in Figure 1. In a two-dimensional space, the shallow water equations with source

terms due to variable bottom topography and bed friction eﬀects can be written as follows:











∂th+∂xhu +∂yhv = 0,

∂thu +∂xhu2+g

2h2+∂yhuv=−gh∂xB−Cfu√u2+v2,

∂thv +∂xhuv+∂yhv2+g

2h2=−gh∂yB−Cfv√u2+v2,

(2.1)

where trepresents the time, xand yare the cartesian coordinates and gis the gravity

acceleration. The source terms in the momentum equations are the contributions of the

bottom topography Band the friction terms Cfu√u2+v2and Cfv√u2+v2, where the bed

roughness coeﬃcient Cfis computed using the Manning formula:

Cf=gn2

h1/3,(2.2)

with nfbeing the Manning coeﬃcient which represents the bed hydraulic resistance to ﬂow.

The frictional resistance may have higher eﬀects on ﬂow velocity as the water depth decreases

since the roughness coeﬃcient Cfincreases by decreasing the water depth [51].

The system of shallow water equations (2.1) can be expressed in the following matrix

form:

∂tU+∂xF(U, B) + ∂yG(U, B) = S(U, B) + M, (2.3)

where we deﬁne new conservative variables U:= [w, qx, qy]Tof the system (2.1) and denote

by w:= B+hthe free-surface elevation and qx:= uh and qy:= vh the water discharges

Figure 1: Schematic of the shallow water model variables on a coastal area.

along the coordinate directions Ox and Oy, respectively. The vectors of ﬂuxes, bottom

topography and friction term (F, G)T,Sand Mrespectively, are deﬁned by:

F:= 





(w−B)+g

2(w−B)2

qxqy

(w−B)







, G := 





qxqy

(w−B)

(w−B)+g

2(w−B)2







S:= 





−g(w−B)∂xB

−g(w−B)∂yB





, M := 





−gn2

h7/3qxqq2

x+q2

−gn2

h7/3qyqq2

x+q2







(2.4)

The Jacobian matrix of the shallow water system (2.3)-(2.4) has the following eigenvalues:

λ1=unx+vny−pgh, λ2=unx+vny,λ3=unx+vny+pgh, (2.5)

which are used in the numerical scheme to compute the one-side local speeds of wave propa-

gation, where n= [nx, ny]Tis the unit normal vector to the cell interfaces of control volumes

used in our numerical methodology in Section 3.

2.2. Stationary steady-state solutions

The shallow water equations (2.1) with source terms is a system of balance laws. Under

some particular initial conditions, this system has trivial steady-state solutions of the “lack

at rest” (u= 0) in the form:

qx≡qy≡0,and w=h+B≡constant. (2.6)

Furthermore, the shallow water system has a nontrivial steady-state solution which corre-

sponds to the situation of steady ﬂow with constant water depth and non-vanishing velocity

over an inclined topography [49]:

h=h0≡constant, qx=q0≡constant, qy≡0, ∂xB=−B0≡constant, and ∂yB≡0,or

h=h0≡constant, qx≡0, qy=q0≡constant, ∂xB≡0,and ∂yB=−B0≡constant.

(2.7)

The expression of the water depth can be obtained by replacing the solution (2.7) in the

momentum equations of the shallow water system (2.1).

h0=n2

fq2

B03/10

.(2.8)

The aforementioned trivial and nontrivial steady-state solutions (2.6)-(2.7) are used to test

the proposed numerical model in terms of well-balanced and accuracy.

3. Numerical methodology

3.1. Numerical scheme

In this section, we brieﬂy describe the ﬁnite volume central-upwind scheme on unstruc-

tured triangular grids applied to solve the system of shallow water equations (2.1) [50]. The

computational domain is partitioned into triangular cells Tjof size |Tj|, and centers of mass

Cj:= (¯xj,¯yj). We denote by (∂Tj)kthe common interface of the cell Tjand its neighboring

cells Tjk of length djk,k= 1,2,3. Let njk := (cos(θjk), sin(θjk)) be the unit vector normal

to (∂Tj)kpointed towards to the cell Tjk, and (xjk, yjk)be the coordinates of the midpoint

Njk of the cell interface (∂Tj)kwith corresponding vertices Njkiof coordinates (xjki, yjki),

i= 1,2as shown in Figure 2.

••

•

Tjk

•

Cjk •

Njk

(∂Tj)k

njk

Njk1

Njk2

Figure 2: Schematic of triangular cells

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Anumericalmodelpreservingnontrivialsteady-statesolutionsforpredictingwavesrun-uponcoastalareasHasanKarjouna,AbdelazizBeljadida;b;aMohammedVIPolytechnicUniversity,GreenCity,MoroccobUniversityofOttawa,Ottawa,CanadaAbstractInthisstudy,anumericalmodelpreservingaclassofnontrivialsteady-statesolutionsisp...

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A numerical model preserving nontrivial steady-state solutions for predicting waves run-up on coastal areas Hasan Karjouna Abdelaziz Beljadidab

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