Adaptive numerical modelling of tsunami wave generation and propagation with FreeFem++

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generation and propagation with FreeFem++

Georges Sadaka, Denys Dutykh

To cite this version:

Georges Sadaka, Denys Dutykh. Adaptive numerical modelling of tsunami wave generation and prop-

agation with FreeFem++. 2020. �hal-02912526�

Adaptive numerical modelling of tsunami wave generation and propagation with FreeFem++

Georges SADAKA 1,†* , Denys DUTYKH 2,†

1 Laboratoire de Mathématiques Raphaël Salem, Université de Rouen Normandie, CNRS UMR 6085, Avenue de l’Université, BP 12, F-76801 Saint-Étienne-du-Rouvray, France; Georges.Sadaka@univ-rouen.fr

2 Univ. Grenoble Alpes, Univ. Savoie Mont Blanc, CNRS, LAMA, 73000 Chambéry, France;

Denys.Dutykh@univ-smb.fr

* Correspondence: Georges.Sadaka@univ-rouen.fr; Tel.: +33(0)781978413 (G.S.)

† These authors contributed equally to this work.

Academic Editor: Anawat Suppasri

Version August 6, 2020 submitted to Geosciences

Abstract:A simplified nonlinear dispersive system of BBM-type, initially derived by D. MITSOTAKIS,

1

is employed here in order to model the generation and propagation of surface water waves over

2

variable bottom. The simplification consists in applying the so-called BOUSSINESQapproximation.

3

Using the finite element method and theFreeFem++software, we solve numerically this system for

4

three different complexities for the bathymetry function: a flat bottom case, a variable bottom in

5

space, and a variable bottom both in space and in time. The last case is illustrated with the JAVA2006

6

tsunami event. This article is designed rather as a tutorial paper even if it contains the description of

7

completely new adaptation techniques.

8

Keywords: tsunami wave; finite elements; mesh adaptation; domain adaptation; co-seismic

9

displacements; tsunami wave energy;FreeFem++; unstructured meshes

10

PACS:47.35.Bb; 02.60.-x

11

MSC:76B15; 65N30; 65N50

12

1. Introduction

13

Tsunami waves represent undeniably a complex natural process. Moreover, they represent a major

14

risk for exposed coastal areas including also the local populations, infrastructure,etc. The present work

15

is devoted to the modelling tsunami generation and propagation processes. Moreover, this article is

16

designed as a tutorial paper in order to show to the readers how easily these processes can be modelled

17

in the framework of theFreeFem++open source finite element software. Traditionally, tsunami waves

18

are modelled using hydrostatic models [1–4]. In the present manuscript we employ a non-hydrostatic

19

BOUSSINESQ-type system to be specified below. This class of models is distinguished by the application

20

of the so-called BOUSSINESQapproximation [5]. They can be used to study a variety of water wave

21

phenomena in harbors, coastal dynamics and, of course, tsunami generation and propagation problems

22

[6–10].

23

In this study we consider a BBM–BBM system derived by D. MITSOTAKISin 2D over a variable bottom in spaceh

(

x,y

)

and in timeζ

(

x,y,t

)

[11]:

ηt

+ ∇ · ((

h

+

η

)

V

) +

ζt

+

A^˜

∇ ·

h²

∇

ζt

+ ∇ ·

ⁿAh²

[∇ (∇

h

·

V

) + ∇

h

∇ ·

V

] −

bh²

∇

ηt

o

=

0 , Vt

+

g

∇

η

+

¹

2

∇|

V

|

²

+

Bgh

[∇ (∇

h

· ∇

η

) + ∇

h∆η

] −

dh²∆Vt

−

Bh

∇

ζtt

=

0 , (1)

Submitted toGeosciences, pages 1 – 24 www.mdpi.com/journal/geosciences

Figure 1.The sketch of the physical domainΩ.

where

ba

=

θ

−

¹ 2

, bb

=

¹ 2

(

θ

−

1

)

²

−

¹ 3

, A˜

=

νba

− (

1

−

ν

)

bb, A

= −

bb, B

=

1

−

θ,

b

=

¹ 2

θ²

−

¹

3

(

1

−

ν

)

, d

=

¹ 2

1

−

θ²

(

1

−

µ

)

and g is the acceleration due to gravity. System (1) is an asymptotic approximation to the

24

three-dimensional full EULERequations describing the irrotational free surface flow of an ideal fluid

25

Ω

⊂

R³[12,13], which is bounded below by

−

z_b

(

x,y,t

) = −

h

(

x,y

) −

ζ

(

x,y,t

)

and above by the free

26

surface elevationη

(

x,y,t

)

(cf. Figure1).

27

The variables in (1) areX

= (

x,y

) ∈

Ωandt>0 are proportional to position along the channel

28

and time, respectively.η

=

η

(

X,t

)

being proportional to the deviation of the free surface departing

29

from its rest position andV

=

V

(

X,t

) =

^u

(

X,t

)

v

(

X,t

)

!

= (

u,v

)

^>

= (

u;v

)

being proportional to the

30

horizontal velocity of the fluid at some height. In our study, we suppose thatη

= O(

a

)

, with the

31

characteristic wave amplitudea(in other words,ηis the difference between the water free surface and

32

the still water level). Also we setλ

= O(`)

be the wave length. In addition, we limit ourselves to the

33

case whereη

+

z_b >0 (there are no dry zones in our computations).

34

This paper is organized as follows. In Section2we present the space and time discretization of

35

Equations (1). In Section3, we present the new domain adaptation technique. In Section4, we establish

36

the convergence of our numerical code, which validates the adequacy of the chosen finite element

37

discretization. Then, with this code we simulate the propagation of a tsunami-like wave generated by

38

the moving bottom (e.g.an earthquake). We present several test cases in various regions of the world.

39

First, we take a MEDITERRANEANsea-shaped computational domain with flat bottom and we solve

40

the sBBM system (1) in it. The mesh in this study is generated from a space image. Then, we consider

41

the JAVAisland region with real world bathymetry. Finally, we apply this solver to simulate a realistic

42

example of a tsunami wave near the JAVAisland which took place in 2006. We note that all numerical

43

simulations were done using theFreeFem++software [14], which is an open source platform to solve

44

Partial Differential Equations (PDEs) numerically, based on Finite Element Methods (FEM). The main

45

conclusions of this study are outlined in Section5.

46

2. Discretization of the BBM–BBM system

47

In this section, we present the spatial discretization of (1) using Finite Element Method (FEM)

48

withP1continuous piecewise linear elements. For the time marching scheme we use an explicit second

49

order RUNGE–KUTTAmethod.

50

2.1. Spatial discretization

51

We letΩ be a convex, plane domain, and

T

_h be a regular, quasi-uniform triangulation of Ω with triangles of maximum sizeh < 1. Setting V_h

= {

v_h

∈

C⁰

(

_Ω^¯

)

;v_h

|

_T

∈

P1

(

T

)

,

∀

T

∈ T

_h

}

be a finite-dimensional, whereP1is the set of all polynomials of degree

≤

1 with real coefficients and denoting by

h·

;

·i

the standardL²inner product onΩ, we consider the weak formulation of System (1):

findη_h,u_h,v_h

∈

V_hsuch that

∀

φ^η_h,φ_h^u,φ^v_h

∈

V_h, we have:

D

η_ht

−

b

∇ ·

h²

∇

η_ht

+ ∇ · ((

h

+

η_h

) (

u_h;v_h

)) +

ζt;φ_h^η

E

+

^DA^˜

∇ ·

h²

∇

ζt

;φ^η_h E

+

^D

∇ ·

Ah²

[∇ (∇

h

· (

u_h;v_h

)) + ∇

h

∇ · (

u_h;v_h

)]

;φ^η_h E

=

0, u_ht

−

dh²∆uht

+

gη_xh

+

u_hu_hx

+

v_hv_hx

−

Bhζxtt;φ_h^u

+

BgD hh

(∇

h

· ∇

η_h

)

_x

+

hx∆ηh

i;φ_h^u E

=

0, D

v_ht

−

dh²∆v_ht

+

gη_yh

+

u_hu_hy

+

v_hv_hy

−

Bhζytt;φ_h^u E

+

BgD hh

(∇

h

· ∇

η_h

)

_y

+

hy∆η_hi

;φ_h^v E

=

0.

(2) For simplicity, we setφ_h^η

=

_Φ^η, φ^u_h

=

_Φ^u, φ^v_h

=

_Φ^v, η_h

= E

, u_h

= U

,v_h

= V

, so that system (2) can be rewritten in the following way:





























 D

∂_t

E −

b

∇ ·

h²

∇

∂_t

E

;Φ^ηE

= −

(

h

+ E)∇ · (U

;

V) + (

hx

+ E

_x

)U + (

hy

+ E

_y

)V +

ζ_t

+

A^˜

∇ ·

h²

∇

ζt

+

A

∇ ·

h²

[∇ (∇

h

· (U

;

V)) + ∇

h

∇ · (U

;

V)]

;Φ^η

= F (E

,

U

,

V

,Φ^η

)

, D

(

I_d

−

dh²∆

)

∂t

U

;Φ^uE

= −

^Dg

E

_x

+ U U

_x

+ V V

_x

+

Bghh

(∇

h

· ∇E )

_x

+

hx∆

E

ⁱ

−

Bhζxtt;Φ^uE

= G (E

,

U

,

V

,Φ^u

)

, D

(

Id

−

dh²∆

)

∂t

V

;Φ^vE

= −

^Dg

E

y

+ U U

y

+ V V

y

+

Bghh

(∇

h

· ∇E )

_y

+

hy∆

E

ⁱ

−

Bhζytt;Φ^vE

= H (E

,

U

,

V

,Φ^u

)

.

(3) After integrating by parts, the left hand side of (3) becomes:

−

^Db

∇ ·

h²

∇

∂t

E

;Φ^ηE

=

bD

h²

∇

∂t

E

;

∇(

_Φ^η

)

^E

−

Z

Γn

bh²Φ^η∂

(

∂t

E )

∂n ∂γ,

−

^Ddh²∆∂t

U

;Φ^uE

=

dD

h²

∇

∂t

U

;

∇

_Φ^uE

+

d

h

2h

∇

h

· ∇

∂t

U

;Φ^u

i −

Z

Γn

dh²Φ^u∂

(

∂_t

U)

∂n ∂γ, and

−

^Ddh²∆∂t

V

;Φ^vE

=

dD

h²

∇

∂t

V

;

∇

_Φ^vE

+

d

h

2h

∇

h

· ∇

∂t

V

;Φ^v

i −

Z

Γn

dh²Φ^v∂

(

∂t

V)

∂n ∂γ,

whereΓnis the boundary of the domainΩ. Dealing with the right-hand side

F (E

,

U

,

V

,Φ^η

)

of the first equation in System (3), we expand the two complex terms which are multiplied byAand ˜Asuch as:

D

∇ ·

h²

∇

ζt

;Φ^ηE

=

h²ζxt

x

+

h²ζyt

y;Φ^η

=

^D2hhxζxt

+

h²ζxxt

+

2hhyζyt

+

h²ζyyt;Φ^ηE , and

D

∇ ·

ⁿh²

[∇ (∇

h

· (U

;

V)) + ∇

h

∇ · (U

;

V)]

^o;Φ^ηE

=

^D

∇ ·

ⁿh²h

hx

U +

hy

V

_x; hx

U +

hy

V

_y

+

hx

∇ · (U

;

V)

;hy

∇ · (U

;

V)

^io;Φ^ηE

=

^D

∇ ·

h²hxx

U +

h²hx

U

_x

+

h²hxy

V +

h²hy

V

_x

+

h²hx

∇ · (U

;

V)

;h²hxy

U +

h²hx

U

_y

+

h²hyy

V +

h²hy

V

_y

+

h²hy

∇ · (U

;

V)

;Φ^ηE

=

^D

(

2hhxhxx

+

2hhyhxy

+

h²hxyy

+

h²hxxx

)U + (

2hhxhxy

+

2hhyhyy

+

h²hyyy

+

h²hxxy

)V + (

4hh²_x

+

3h²hxx

+

2hh²_y

+

h²hyy

)U

_x

+

2

(

h²hxy

+

hhxhy

)U

_y

+ (

4hh²_y

+

3h²hyy

+

2hh²_x

+

h²hxx

)V

y

+

2

(

hhxhy

+

h²hxy

)V

x;Φ^ηE

+

^D2h²hx

U

xx;Φ^ηE

+

^Dh²hy

U

_xy;Φ^ηE

+

^Dh²hx

U

_yy;Φ^ηE

+

^Dh²hy

V

_xx;Φ^ηE

+

^Dh²hx

V

_xy;Φ^ηE

+

^D2h²hy

V

_yy;Φ^ηE . On the other hand, we have:

D2h²hx

U

_xx;Φ^ηE

= −

^D2h²hx

U

_x;Φ^ηx

E

−

^D

(

4hh²_x

+

2h²hxx

)U

_x;Φ^ηE

+

Z

Γn

2h²hxΦ^η∂

U

∂n∂γ, Dh²hy

U

_xy;Φ^ηE

= −

^Dh²hy

U

_x;Φ^ηy

E

−

^D

(

2hh²_y

+

h²hyy

)U

_x;Φ^ηE

+

Z

Γn

h²hyΦ^η∂

U

∂n∂γ, Dh²hx

U

_yy;Φ^ηE

= −

^Dh²hx

U

_y;Φ^ηy

E

−

^D

(

2hhxhy

+

h²hxy

)U

_y;Φ^ηE

+

Z

Γn

h²hxΦ^η∂

U

∂n∂γ, Dh²hy

V

xx;Φ^ηE

= −

^Dh²hy

V

x;Φ^ηx

E

−

^D

(

2hhxhy

+

h²hxy

)V

x;Φ^ηE

+

Z

Γn

h²hyΦ^η∂

V

∂n∂γ, D

h²hx

V

xy;Φ^ηE

= −

^Dh²hx

V

x;Φ^ηy

E

−

^D

(

2hhxhy

+

h²hxy

)V

x;Φ^ηE

+

Z

Γn

h²hxΦ^η∂

V

∂n∂γ, D

2h²hy

V

_yy;Φ^ηE

= −

^D2h²hy

V

_y;Φ^ηy

E

−

^D

(

4hh²_y

+

2h²hyy

)V

_y;Φ^ηE

+

Z

Γn

2h²hyΦ^η∂

V

∂n∂γ, and, consequently, we deduce the final form of

F (E

,

U

,

V

,Φ^η

)

as follows:

F (E

,

U

,

V

,Φ^η

) = −

(

h

+ E )∇ · (U

;

V) + (

hx

+ E

_x

)U + (

hy

+ E

_y

)V +

ζ_t;Φ^η

−

A^˜D

2hhxζxt

+

h²ζxxt

+

2hhyζyt

+

h²ζyyt;Φ^ηE

−

AD

(

2hhxhxx

+

2hhyhxy

+

h²hxyy

+

h²hxxx

)U + (

2hhxhxy

+

2hhyhyy

+

h²hyyy

+

h²hxxy

)V +

h²hxx

U

_x

+

h²hxy

U

_y

−

2hhxhy

V

_x

+(

h²hyy

+

2hh²_x

+

h²hxx

)V

_y;Φ^ηE

+

AD

2h²hx

U

_x

+

h²hy

V

_x;Φ^ηx

E

+

^Dh²hy

U

_x

+

h²hx

U

_y

+

h²hx

V

_x

+

2h²hy

V

_y;Φ^ηy

E

−

A Z

Γn

(

3h²hx

+

h²hy

)

_Φ^η∂

U

∂n

+ (

h²hx

+

3h²hy

)

_Φ^η∂

V

∂n

∂γ.

For the right-hand side

G (E

,

U

,

V

,Φ^u

)

of the second equation in System (3), we have:

G (E

,

U

,

V

,Φ^u

) = −

g

E

x

+ U U

x

+ V V

x

+

Bgh

hx

E

x

+

hy

E

y

x

+

hx

(E

xx

+ E

yy

)

−

Bhζxtt;Φ^u

= −

g

E

_x

+ U U

_x

+ V V

_x

+

Bg hhxx

E

_x

+

hhxy

E

_y

−

Bhζxtt;Φ^u

−

Bg

2hhx

E

_xx

+

hhy

E

_xy

+

hhx

E

_yy;Φ^u

= −

g

E

_x

+ U U

_x

+ V V

_x

+

Bg hhxx

E

_x

+

hhxy

E

_y

−

Bhζxtt;Φ^u

+

Bg

h

2hhx

E

_x;Φ^ux

i +

BgD

(

2h²_x

+

2hhxx

)E

_x;Φ^uE

+

BgD

hhy

E

_x

+

hhx

E

_y;Φ^u_yE

+

BgD

(

h²_y

+

hhyy

)E

_x

+ (

hxhy

+

hhxy

)E

_y;Φ^uE

−

Z

Γn

Bg

(

3hhx

+

hhy

)

_Φ^u∂

E

∂n∂γ

= −

^Dg

Id

−

B

hhxx

+

2h²_x

+

hhyy

+

h²_y

E

x

+ U U

x

+ V V

x

−

Bghxhy

E

y

−

Bhζxtt;Φ^uE

+

Bg

h

2hhx

E

x;Φ^u_x

i

+

BgD

hhy

E

_x

+

hhx

E

_y;Φ^uy

E

−

Z

Γn

Bg

(

3hhx

+

hhy

)

_Φ^u∂

E

∂n∂γ.

Finally, for the right-hand side

H (E

,

U

,

V

,Φ^v

)

of the third equation in System3, we have:

H (E

,

U

,

V

,Φ^v

) = −

^Dg

E

y

+ U U

y

+ V V

y

+

Bghh

hx

E

x

+

hy

E

y

y

+

hy

(E

xx

+ E

yy

)

ⁱ

−

Bhζytt;Φ^vE

= −

g

E

_y

+ U U

_y

+ V V

_y

+

Bg hhxy

E

_x

+

hhyy

E

_y

−

Bhζytt;Φ^v

−

Bg

hhy

E

_xx

+

hhx

E

_xy

+

2hhy

E

_yy

)

;Φ^v

= −

g

E

_y

+ U U

_y

+ V V

_y

+

Bg hhxy

E

_x

+

hhyy

E

_y

−

Bhζytt;Φ^v

+

Bg

hhy

E

_x;Φ^v_x

+

Bg

(

hxhy

+

hhxy

)E

_x;Φ^v

+

BgD

hhx

E

_x

+

2hhy

E

_y;Φ^v_yE

+

BgD

(

hxhy

+

hhxy

)E

_x

+ (

2h²_y

+

2hhyy

)E

_y;Φ^vE

−

Z

Γn

Bg

(

hhx

+

3hhy

)

Φ^v∂

E

∂n∂γ

= −

^D

−

Bg

(

2hxhy

+

hhxy

)E

_x

+ U U

_y

+ V V

_y

+

g

I_d

−

B

hhyy

−

2h²_y

E

_y

−

Bhζytt;Φ^vE

+

Bg

hhy

E

_x;Φ^vx

+

BgD

hhx

E

x

+

2hhy

E

y;Φ^v_yE

−

Z

Γn

Bg

(

hhx

+

3hhy

)

_Φ^v∂

E

∂n∂γ.

However, the model presented above contains some drawbacks. In particular, when the bathymetry function contains steep gradients, it causes instabilities in the numerical solution. We have to mention that this problem is well-known in the framework of BOUSSINESQ-type equations [15]. In order to avoid this kind of problems and to have a robust numerical model, we take two measures.

First of all, we perform the smoothing of the bathymetry data which is fed into the model. In this way, we avoid noise in the bathymetry gradient. As a second and more radical step, we neglect higher order derivatives of the bathymetry function as it was proposed earlier in [11]. Thus, from now on we shall use the following system of equations:



















D

∂t

E

;Φ^ηE

+

b

h²

∇

∂t

E

;

∇(

Φ^η

)

−

Z

Γn

bh²Φ^η∂

(

∂t

E )

∂n ∂γ

= F (E

,

U

,

V

,Φ^η

)

D

∂_t

U

;Φ^uE

+

d

h²

∇

∂_t

U

;

∇

Φ^u

+

d

h

2h

∇

h

· ∇

∂_t

U

;Φ^u

i −

Z

Γn

dh²Φ^u∂

(

∂t

U )

∂n ∂γ

= G (E

,

U

,

V

,Φ^u

)

D

∂t

V

;Φ^vE

+

d

h²

∇

∂t

V

;

∇

_Φ^v

+

d

h

2h

∇

h

· ∇

∂t

V

;Φ^v

i −

Z

Γn

dh²Φ^v∂

(

∂t

V)

∂n ∂γ

= H (E

,

U

,

V

,Φ^v

)

(4) with

F (E

,

U

,

V

,Φ^η

) = −

(

h

+ E )∇ · (U

;

V) + (

hx

+ E

_x

)U + (

hy

+ E

_y

)V +

ζt;Φ^η

−

A^˜

2hhxζxt

+

2hhyζyt;Φ^η

−

AD

−

2hhxhy

V

_x

+

2hh²_x

V

_y;Φ^ηE

+

AD

2h²hx

U

_x

+

h²hy

V

_x;Φ^ηx

E

+

^Dh²hy

U

_x

+

h²hx

U

_y

+

h²hx

V

_x

+

2h²hy

V

_y;Φ^ηy

E

−

A Z

Γn

(

3h²hx

+

h²hy

)

_Φ^η∂

U

∂n

+ (

h²hx

+

3h²hy

)

_Φ^η∂

V

∂n

∂γ,

G (E

,

U

,

V

,Φ^u

) = −

^Dg

I_d

−

B

2h²_x

+

h²_y

E

_x

+ U U

_x

+ V V

_x

−

Bghxhy

E

_y

−

Bhζxtt;Φ^uE

+

Bg

h

2hhx

E

_x;Φ^ux

i +

BgD

hhy

E

x

+

hhx

E

y;Φ^u_yE

−

Z

Γn

Bg

(

3hhx

+

hhy

)

_Φ^u∂

E

∂n∂γ, and

H (E

,

U

,

V

,Φ^v

) = −

^D

−

2Bghxhy

E

x

+ U U

y

+ V V

y

+

g

Id

−

2Bh²_y

E

y

−

Bhζytt;Φ^vE

+

Bg

hhy

E

x;Φ^v_x

+

BgD

hhx

E

x

+

2hhy

E

y;Φ^v_yE

−

Z

Γn

Bg

(

hhx

+

3hhy

)

Φ^v∂

E

∂n∂γ.

2.2. Time marching scheme

52

Our method is based on the explicit second order RUNGE–KUTTA scheme. For that, let us denote by

(E

ⁿ⁺¹,

U

ⁿ⁺¹,

V

ⁿ⁺¹

)

_and

(E

ⁿ,

U

ⁿ,

V

ⁿ

)

the approximate values at timet

=

tⁿ⁺¹andt

=

tⁿ, respectively and byδtthe time step size. Then, owing to (4), the unknown fields at timet

=

tⁿ⁺¹are defined as the solution of the following system:















hE

ⁿ⁺¹;Φ^η

i = hE

ⁿ

+ E

^k1

+ E

^k2 2 ;Φ^η

i

,

hU

ⁿ⁺¹;Φ^u

i = hU

ⁿ

+ U

^k1

+ U

^k2

2 ;Φ^u

i

,

hV

ⁿ⁺¹;Φ^v

i = hV

ⁿ

+ V

^k1

+ V

^k2

2 ;Φ^v

i

,

(5)

where

D

E

^k1;Φ^ηE

+

bD

h²

∇E

^k1;

∇(

Φ^η

)

^E

−

Z

Γn

bh²Φ^η∂

(E

^k1

)

∂n ∂γ

=

δt

· F (E

ⁿ,

U

ⁿ,

V

ⁿ,Φ^η

)

, D

U

^k1

+

2dh

∇

h

· ∇U

^k1;Φ^uE

+

dD

h²

∇U

^k1;

∇

_Φ^uE

−

Z

Γn

dh²Φ^u∂

(U

^k1

)

∂n ∂γ

=

δt

· G (E

ⁿ,

U

ⁿ,

V

ⁿ,Φ^u

)

, D

V

^k1

+

_2dh

∇

h

· ∇V

^k1;Φ^vE

+

_d^D_h²

∇V

^k1;

∇

Φ^vE

−

Z

Γn

dh²Φ^v∂

(V

^k1

)

∂n ∂γ

=

_δt

· H (E

ⁿ,

U

ⁿ,

V

ⁿ,Φ^v

)

(6) and

D

E

^k2;Φ^ηE

+

bD

h²

∇E

^k2;

∇(

Φ^η

)

^E

−

Z

Γn

bh²Φ^η∂

(E

^k2

)

∂n ∂γ

=

δt

· F

E

ⁿ

+ E

^k1,

U

ⁿ

+ U

^k1,

V

ⁿ

+ V

^k1,Φ^η , D

U

^k2

+

2dh

∇

h

· ∇U

^k2;Φ^uE

+

dD

h²

∇U

^k2;

∇

_Φ^uE

−

Z

Γn

dh²Φ^u∂

(U

^k2

)

∂n ∂γ

=

δt

· G

E

ⁿ

+ E

^k1,

U

ⁿ

+ U

^k1,

V

ⁿ

+ V

^k1,Φ^u , D

V

^k2

+

2dh

∇

h

· ∇V

^k2;Φ^vE

+

dD

h²

∇V

^k2;

∇

Φ^vE

−

Z

Γn

dD²Φ^v∂

(V

^k2

)

∂n ∂γ

=

δt

· H

E

ⁿ

+ E

^k1,

U

ⁿ

+ U

^k1,

V

ⁿ

+ V

^k1,Φ^v . (7) 3. New domain adaptation, domains computation and initial data

53

We present here the new domain adaptation technique that will be compared in the sequel with

54

the mesh adaptation used inFreeFem++.

55

3.1. New domain adaptation technique

56

Since some computation domains for many applications (here forTsunamiwaves) may be huge

57

and the initial data is concentrated in a small domain, a circle

C (

O,R

)

or a rectangle

[

a,b

] × [

c,d

]

,

58

before starting to propagate in the domain, we present here an idea to build a moving computation

59

domain around the solution only, as when we use a mesh adaptation. The difference between these

60

two methods is that the moving domain will be a cut from the initial one;i.e.all initials vertices, edges

61

and boundary labels are conserved and a new label is defined for the new boundary; while the mesh

62

adaptation technique don’t conserve the initials vertices and edges, so when we make interpolation of

63

solution from old to new mesh we will lose some information in the mesh adaptation technique but

64

not with the moving domain.

65

Firstly, we cut from the initial meshThinita circle or a rectangle zoneThwhere our initial solution

66

lives (usingtruncinFreeFem++), secondly we compute the initial solutionu0and we interpolate it to

67

uadapt

∈

P1finite element (usinginterpolateinFreeFem++), and for each adaptation we follow this

68

algorithm:

69

• We deduce the limit min max ofThonxandydirection (usingboundingboxinFreeFem++).

70

• We addepsadaptfrom each side in order to build the new rectangleTh1(cutted fromThinit)

71

that containsTh(usingtruncinFreeFem++).

72

(a) (b)

Figure 2.Left (a): the mesh aroundCRETEisland. Right (b): the place of: wave gauge and?: epicenter.

• We interpolateuadapt

∈

Thtouadapt1

∈

Th1(usinginterpolateinFreeFem++).

73

• We smooth the function obtained fromabs(uadapt1)

≥

erradaptusing:

βu

−

_∆u

=

βf, (8)

wheref

= (|

uadapt1

| ≥

erradapt

)

, with zero DIRICHLETBC only on the new boundary label

74

ofTh1and a NEUMANNBC in the other boundary label.

75

• We cut fromTh1, respecting tou > isoadapt, the final meshThnew(usingtruncinFreeFem++).

76

Finally, we replaceThby the new meshThnewand we interpolate the solution from the old mesh to

77

the new one (usinginterpolateinFreeFem++). We use a reflective Boundary Condition (BC) on the

78

new boundary,i.e. zero NEUMANNBC forηand zero DIRICHLETBC forV, cause our BBM–BBM

79

system gives artificial numerical explosion on the boundary if we do not use any BC or if we use only

80

NEUMANNBC forηandV.

81

For the BBM–BBM system over a flat bottom, we use a mesh generated through a photo of the

82

MEDITERRANEANsea (a cut of the mesh around the CRETEisland is shown in Figure2at left panel)

83

and for the BBM–BBM system over a variable bottom in space and in time, we use a mesh generated

84

using an imported bathymetry fxyfor the sea near the JAVAisland which can be downloaded from

85

the NOAA¹website where in this case, the mesh generated is for the area, where the amplitude is

86

zero. We can smooth the bathymetric data obtained from NOAA (cf.Figure3, left panel) by solving (8)

87

withf

=

fxy. For all simulations with realistic bathymetry, we useβ

=

20 in (8) to smooth the initial

88

bathymetry after the generation of the mesh (cf.Figure3, right panel) in order to ensure the stability

89

of the numerical method, we also note that in order to be in a big deep water wave regime for the

90

BBM–BBM system we change the depth close to the shoreline to 100m.

91

The bathymetry data downloaded from the NOAA website are in degree coordinate and we need to convert them to meters. So, on the first hand, we must know the degree of Latitude (South and North) and of Longitude (West and East) of our domain where we can deduce the Latitude lat0

=

.5

(

lat_South

+

lat_North

)

and the Longitudelong0

=

.5

(

long_West

+

longEast

)

. On the other hand, we must take into account the spherical shape of the EARTH, even if it does not play significant role because of the small spatial scale of the experiments. So, we know that the radius of the EARTHnear

1 https://maps.ngdc.noaa.gov/viewers/wcs-client/

(a) (b)

Figure 3.Left (a): Bathymetry downloaded from the NOAA website, (min=−7239mand max=3002m).

Right (b): smoothed bathymetry withβ=20in(8), (min=−6207mand max=−100m).

the equator isRequator

=

6378, 137km, and near to the poleR_pole

=

6356, 752km, thus the radius of our domain equals to:

R

=

v u u u t

R²_equatorcos

(

lat0

·

π/180

)

²

+

R²_polesin

(

lat0

·

π/180

)

² Requatorcos

(

lat0

·

π/180

)

²

+

Rpolesin

(

lat0

·

π/180

)

^2. So, we move the mesh of our domain using the following translation (coefl0

=

πR/180):

[

x;y

] −→ [(

x

−

lon0

)

_cos

(

πy/180

)

_coefl0;

(

y

−

lat0

)

_coefl0

]

_. 3.2. Initial data

92

Tsunami waves considered in this study are generated by the co-seismic deformation of the Ocean’s or sea’s bottom due to an earthquake. The adopted modelling of the tsunami wave generation process is inspired by [8,11,16,17]. The co-seismic displacement is computed according to the celebrated OKADA’s solution [18,19]. We assume the dip-slip dislocation process underlying the earthquake.

The vertical component of displacement vector

O(

x,y

)

is given by the following formulas employing CHINNERY’s notation,cf.[16,17]:

f

(

ξ,η

) ||=

f

(

ξ,p

) −

f

(

ξ,p

−

W

) −

f

(

ξ

−

L,p

) +

f

(

ξ

−

L,p

−

W

)

,

O (

x,y

) = −

^U

2π

dq˜

R

(

R

+

ξ

) +

sinδarctanξη

qR

−

Isinδcosδ

, where

ξ

= (

x

−

x0

)

cosφ

+ (

y

−

y0

)

sinφ, Y

= −(

x

−

x0

)

sinφ

+ (

y

−

y0

)

cosφ, p

=

Ycosδ

+

dsinδ, q

=

Ysinδ

−

dcosδ,

˜

y

=

ηcosδ

+

qsinδ, ˜d

=

ηsinδ

−

qcosδ, R²

=

ξ²

+

η²

+

q²

=

ξ²

+

y˜²

+

d^˜2, X²

=

ξ²

+

q²

(a) (b)

Figure 4.Geometry of the source model (left) and the initial solution forη(right, min=−0.46m, max=0.71 m).

and

I

=









 µ λ

+

µ

2

cosδarctanη

(

X

+

qcosδ

) +

X

(

R

+

X

)

sinδ

ξ

(

R

+

X

)

cosδ if cosδ

6=

0, µ

λ

+

µ ξsinδ

R

+

d^˜ if cosδ

=

_0.

Here,WandLare the width and the length of the rectangular fault,

(

x,y

)

are the points where we

93

computes displacements,

(

x0,y0

)

is the epicenter,d

=

fault depth

(

x0,y0

) +

Wsinδ,δis the dip angle,

94

θis the rake angle,Dis the BURGERS’s vector,U

= |

D

|

sinθis the slip on the fault,φis the strike angle

95

which is measured conventionally in the counter-clockwise direction from the North (cf. Figure4(left)),

96

µ, λare the LAMÉconstants derived from elastic-wave velocities:λ

=

ρc V_P²

−

V_S²

andµ

=

ρcV_S²,

97

whereρ_cis the crust density,VPis the compressional-wave (P

−

wave) velocity,VSis the shear-wave

98

(S

−

wave) velocity. TheMatlabscript to compute the OKADAsolution can be downloaded at the

99

following URL:

100

https://mathworks.com/matlabcentral/fileexchange/39819-okada-solution/

101

We shall distinguish here the two types of tsunami wave generation mechanisms [20,21]:active

102

andpassivegeneration mechanisms.

103

3.2.1. Passive generation

104

We remind that the passive generation approach consists in transposing the bottom deformation

105

on the free surface as an initial condition for tsunami propagation codes. In order to compute the initial

106

data forη

(

x,y, 0

) = O(

x,y

)

in meters (cf. Figure4(right)),V

(

x,y, 0

) =

0 which is referred to as apassive

107

generation of a tsunami wave near the JAVAisland, using our domain adaptive technique, we will use

108

the fact that the solution is concentrated in the small rectangle

[

x0

−

3.2W;x0

+

1.2W

] × [

y0

−

L;y0

+

L

]

109

whereL

=

100km,W

=

50km,δ

=

10.35^◦,φ

=

288.94^◦,θ

=

95^◦,U

=

2m,ρc

=

2700kg/m³,V_P

=

6000

110

m/s,VS

=

3400m/s,

(

x0;y0

) = (

107.345^◦,

−

9.295^◦

)

and the fault depth 10km. All these geophysical

111

parameters can be downloaded from this file hosted by USGS:

112

https://Earthquake.usgs.gov/archive/product/finite-fault/usp000ensm/us/1486510367579/

113

web/p000ensm.param

114

3.2.2. Active generation

115

In contrast to passive generation, the active generation approach consists in generating a tsunami waves by computing fluid layer interaction with moving bottom. For a more realistic case of the JAVA

2006 event, we use precisely this so-calledactivegeneration approach by following [8,22]. In this case we consider zero initial conditions for both the free surface elevation and the velocity field, and assume

(a) (b)

Figure 5.Left panel (a): Surface projection of the fault’s plane and the mesh around,: wave gauge,?: epicenter.

Right panel (b): the14-th Okada solution (min=−0.09m, max=0.17m ).

that the bottom is moving in time. This case may be described by considering the bottom motion formula:

−

z_b

(

x,y,t

) = −

h

(

x,y

) −

ζ

(

x,y,t

)

_with

ζ

(

x,y,t

) =

Nx·N_y i=1

∑

H(

t

−

t_i

) ·

1

−

e^{−α(t−ti)}

· O

_i

(

x,y

)

,

whereNxsub-faults along strike andNysub-faults down the dip angle,

H(

t

)

is the Heaviside step

116

function andα

=

_log

(

₃

)

_{/tr, wheretr}

=

8 s is the rise time. We choose here an exponential scenario,

117

but in practice, various scenarios could be used (instantaneous, linear, trigonometric,etc.) and could

118

be found in [8,16,17,22,23]. Parameters such as sub-fault location

(

x_i,y_i

)

, depthd_i, slipUand rake

119

angleθfor each segment are given in [8, Table 3]. In this table, we notice that the fault’s surface is

120

conventionally divided intoNx

=

21 sub-faults along strike andNy

=

7 sub-faults down the dip angle,

121

leading to a total number ofNx

×

Ny

=

147 equal segments.

122

For our special domain adaptivity technique, since the fault plane is considered to be the

123

rectangle with vertices located at (109.20508^◦(Lon),

−

10.37387^◦(Lat)), (106.50434^◦(Lon),

−

9.45925^◦

124

(Lat)), (106.72382^◦(Lon),

−

8.82807^◦ (Lat)) and (109.42455^◦(Lon),

−

9.74269^◦(Lat)), we will consider

125

that our bottom displacement is concentrated on the big rectangle which is equidistant of 1^◦ from

126

each side of the initial fault plane as in Figure5(left panel), then we compute each OKADAsolution

127

O

_ion a circle of center

(

x_i

−

10m,y_i

−

10m

)

and of radius 6 max

(

L,W

)

and at the end all the OKADA

128

solution will be interpolated on the big rectangle before starting to compute the vertical displacement

129

of the bottomζ

(

x,y,t

)

, in Figure5(right panel) we plot

O

₁₄. For the computation ofζ

(

x,y,t

)

, we

130

start the mesh by a circle of center

(

xc

−

5m,yc

−

5m

)

and of radius 4 max

(

L,W

)

and we adapt the

131

mesh each 3 iterationsi.e.each 6 s by using the following value for the domain adaptationuadapt

=

_ζ,

132

isoadapt=5e-2,erradapt=1e-4,β=5e-9,epsadapt=50e3.

133

We show in Figure6, the bottom displacementζ

(

x,y,t

)

at timet

=

100sandt

=

270susing our

134

domain adaptation technique. We note that after building the OKADAsolution

O(

x,y

)

in thepassive

135

generation or

O

_i

(

x,y

)

in the active generation, we can remark that this solution is non-local and decays

136

slowly to zero, that is why in our domain adaptation technique we put 0 where the absolute value

137

of the solution is less then min

(|

min

(O

_i

(

x,y

))|

,

|

max

(O

_i

(

x,y

))|)

< 9.2m. We make the same thing

138

without adaptive mesh in order to compare the solution using the same initial data.

139

(a) (b)

Figure 6.Bottom displacement at t =100s(left, min=−0.18m, max=0.38m) and at t =270s(right, min=−0.18m, max=0.45m).

4. Numerical simulations

140

In this section, we study first the rate of convergence of our schemes for the BBM–BBM System (4)

141

with non-dimensional and unscaled variablesi.e., withg

=

1 over a variable bottom in space, which

142

establishes the adequacy of the chosen finite element discretization and the used time marching scheme,

143

for the flat bottom case, we refer to [24], where we use the same technique as in this paper. Then,

144

we simulate the propagation of a wave, that is similar to a real-world tsunami wave generated by

145

an earthquake, in the MEDITERRANEANsea with the BBM–BBM model over a flat bottom. Then, we

146

switch to the JAVAisland region with real variable bottom in space. Finally, we study the active tsunami

147

generation scenario which took place in 2006 near the JAVAisland. In all numerical simulations we

148

usedP1continuous piecewise linear functions forη,u,v,handζ.

149

4.1. Rate of convergence

150

We present the evidence here, following the work done for the 1D case of the BBM–BBM system in [25], that the second order RUNGE–KUTTAtime scheme considered for the BBM–BBM variable bottom in space is of order 2. We note that the functionζ

(

x,y,t

)

is only used for the generation of tsunami wave and, thus, will not be taken into account in the convergence rate test. In this example, we take bi-periodic Boundary Conditions (BC) forη_h,u_handv_hon the whole boundary of the square

[

0, 2L

] × [

0, 2L

]

, whereL

=

50 and we consider the following exact solutions:

η_ex

=

.2 cos

(

2πx/L

−

t

)

cos

(

2πy/L

−

t

)

,uex

=

.5 sin

(

2πx/L

−

t

)

cos

(

2πy/L

−

t

)

, vex

=

.5 cos

(

2πx/L

−

t

)

sin

(

2πy/L

−

t

)

, h

(

x,y

) =

1

−

.5 cos

(

2πx/L

)

cos

(

2πy/L

)

, adding an appropriate function to the right-hand side to make these solutions exact. We measure at timeT

=

1 and forθ²

=

²

3, δt

=

^0.01

2ⁿ andδx

=

^2L N

=

^2L

2ⁿ⁺⁵

∀

n

∈ {

0, 1, 2, 3, 4

}

, the following errors N_L2

(

η

) = k

η_h

−

ηex

k

_L2,N_H1

(

η

) = k

η_h

−

ηex

k

_H1,N_L2

(

V

) = k

u_h

−

uex

k

_L2

+ k

v_h

−

vex

k

_L2

N_H1

(

V

) = k

u_h

−

uex

k

_H1

+ k

v_h

−

vex

k

_H1

and we end up with the results reported in Table1. So, theL²rates forηandVis of order

∼

2 and the

151

H¹rates forηandVis of order

∼

1 as shown in the Figure7and which confirms the convergence of

152

the second-order RUNGE–KUTTAscheme in time for the BBM–BBM system with variable bottom in

153

space.

154