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A Novel Scheme of Dispersive Propagation Model for
Earthquake Generated Tsunami
A Laouar, A Guerziz, A Boussaha
To cite this version:
A Novel Scheme of Dispersive Propagation
Model for Earthquake Generated Tsunami
A. Laouar
1;, A. Guerziz
2, A. Boussaha
1, Mous
1 1Department of Mathematics, Badji Mokhtar University
of Annaba, P. O.Box 12, 23000 Annaba, Algeria.
e-mail: abdelhamid.laouar@univ-annaba.dz
e-mail:boussahaaicha@yahoo.fr
2
Department of Physics, Badji Mokhtar University of Annaba,
P. O. Box 12, 23000 Annaba, Algeria.
e-mail: allaoua.guerziz@univ-annaba.org
Abstract
Keywords: Alternating directions scheme; Numerical dispersion-correction; Shallow water.
1
Introduction
2
Phenomenon Description and hypothesises
Figure 1. Tsunami elevation sea
Denoting by the free surface with z = (x; y; t) and h (x; y) the depth of water. - The incompressibility condition
div u = 0; (1)
- The irrotational ‡ow
u= r : (2)
where potential of the velocity vector u = (u; v; w)t: - kinematic condition @ @t + r :r = 0 with z = (x; y; t) : (3) - Dynamic conditions @ @t + 1 2jr j 2 + g = 0 with z = (x; y; t) : (4)
- The impermeability of the bottom
r :rh = 0 with z = h (x; y) : (5)
2.1
Some Models
@ @t + @M @x + @N @y = 0; (6) @M @t + gh @ @x = 0; (7) @N @t + gh @ @y = 0; (8)
where M (= uh) and N ( = vh) are the horizontal discharge ‡ow along the x and y directions. Such that h is a large enough constant depth , u and v are the average speeds calculated on the depth h of the horizontal directions x and y respectively
Remark 1 The numerical solution of the system (6) - (8) lacks precisely be-cause of the neglect of the physical dispersion. Since this system does not take into account the Coriolis acceleration and is therefore suitable for the propaga-tion of the tsunami in short distances in a limited time (t hours)
The second model is the classical linear Boussinesq equations (CLBqs) in-cluding the coriolis force [1] and describing the propagation of distant tsunami:
@ @t + @M @x + @N @y = 0; (9) @M @t + gh @ @x = h2 3 @3M @t@x2+ @3N @t@x@y ; (10) @N @t + gh @ @y = h2 3 @3N @t@y2 + @3M @t@x@y : (11)
The right-hand side terms of Eqs.(10)-(11) represent the frequency dispersion. Remark 2 Note that the system (9)-(11) re‡ects the acceleration Coriolis and describes the spread of a distant tsunami. But the presence of terms of physical dispersion represent downside for the numerical solution. Thus, the convergence of the used algorithms is relatively slow (see [ 6] ).
@ @t + @M @x + @N @y = 0: (12) @M @t + gh @ @x = B + 1 3 h 2 @3M @t @x2 + @3N @t @x @y + Bgh3 @ 3 @x3+ @2 @x @y2 : (13) @N @t + gh @ @y = B + 1 3 h 2 @3N @t @y2 + @3M @t @x @y + Bgh3 @ 3 @y3 + @2 @x2 @y : (14)
Where B is a curve …tting parameter (see, [12]). The system (12)-(14) is called the improved Boussinesq equations (IBqs).
Free surface equation: substitution from (13)-(14) into (12) and eliminate M and N , yields: @2 @t2 gh @2 @x2 + @2 @y2 = Bgh 3 @4 @x4 + 2 @4 @x2@y2 + @4 @y4 ; +h2 B +1 3 @4 @t2@x2+ @4 @t2 @y2 : (15)
Eq.(15) is called ILBq. The frequency dispersion given by the right-hand side terms of eq.(15) may cause serious numerical di¢ culty in practice because of higher order derivatives. An alternative way is to solve a set of lower order partial di¤erential equations, it is the LSWqs. The numerical dispersion induced by the numerical scheme can be manipulated to represent the physical frequency dispersion of the ILBq.(15).
3
Mathematical Formulation and Discretization
of the Problem
Following the approach suggested by Warming and Hyett in [16] the Taylor series expansions of the variables , M and N at the point represented by (k; i; j) are applied to Eqs.(7) and (8). Using the …nite di¤erence method, where the domain is discretized in a regular grid x ywith a …nite number of nodes
spaced x and y,the time axis is discretized in regular steps t, and the derivatives are replaced by di¤erences over small intervals. The equations (7) and (8) discretized by the scheme of ADI method [9] give (see …gure 2.):
Mk+12 i+1 2;j Mk i+1 2;j t 2 + gh k+1 2 i+1;j k+1 2 i;j x = o ( t; x) ; (16) Ni;j+k+11 2 Nk+ 1 2 i;j+1 2 t 2 + gh k+1 i;j+1 k+1 i;j y = o ( t; y) : (17) Mk+12 i+1 2;j Mk i+1 2;j t 2 = @M xi+1 2; yj; tk @t + t 2 @2M x i+1 2; yj; tk @t2 +; +( t) 2 24 @3M x i+1 2; yj; tk @t3 + o t 3 ; (18) k+1 2 i+1;j k+1 2 i;j x = @ xi; yj; tk+1 2 @x + x 2 @2 x i; yj; tk+1 2 @x2 +; +( x) 2 6 @3 x i; yj; tk+1 2 @x3 + o x 3 : (19)
In order to transform the system of equations (6)-(8) in a system similar to that of improved Boussinesq system (12)-(14) and to make a comparison between them , we will proceed as follows: we replace (18) and (19) in (16), and the equation (7) is used to get the term @
3
@x @t2; afterwards, we apply the convection
equation
@M
@t c
@M
@x = 0; (20)
where c =pgh is the speed. To obtain @
3
@x @y2 instead of the term
@3M
@t3 one hand, and on the other hand,
to eliminate the terms @
2M
@t2 and
@2
@x2 because these terms do not appear in
3.1
A New Modi…ed Scheme for Linear Shallow Water
For inviscid ‡uid, the new system of equations can be written as follows: @ @t + @M @x + @N @y = o ( t; x; y) : (21) @M @t + gh @ @x+ gh 2 1 2 @3 @x3 + 2 3 @2 @x @y2 + 3 @2 @t2 @x ; = o t3; x3; x y2; t2 x : (22) @N @t + gh @ @y+ gh 2 1 2 @3 @y3 + 2 3 @2 @x2 @y+ 3 @2 @t2 @y ; = o t3; y3; x2 y; t2 y : (23)
3.2
Free Surface Equation
@2 @t2 c 2 @2 @x2 + @2 @y2 c 2( x) 2 12 1 + 1 C 2 r @4 @x4 + 2 @4 @x2 @y2 + @4 @y4 ; + (1 + 1 2) c2 ( x)2 6 @4 @x2 @y2 c 2( t) 2 4 (1 3) @4 @x2 @t2 + @4 @y2 @t2 ; = O x3; x2 t; t2 x; t3 : (24)
where Cr= c t= x is called Courant-Friedrichs-Lewis number:
Initial and Boundary Conditions @M @t c @M @x = 0; on @ (t) ; (25) @N @t c @N @y = 0; on @ (t) : (26) (x; y; 0) = h; (27) M (x; y; 0) = N (x; y; 0) = 0: (28)
3.3
Determination of Dispersion-Correction parameters
Comparing Eq.(24)with the ILBq (15), these equations are seen to be identical as long as the following relations are satis…ed:
1= Cr2 12Bh2 x2 1; (29) 2= 1+ 1; (30) and 3= 1 B +13 h2 c2 t2 : (31)
The value of B is not limited to the values discussed in [11]. To verify the stability condition of the current ADI scheme, the value of B can be chosen as
(C2 r 2) x2 12h2 B g t2 12h (32)
The following relation is obtained if B is unique
( x)2= 4h2 2gh( t)2: (33)
3.4
A New Modi…ed Scheme for Weakly Viscous ‡uid
Now we are going to introduce weak dissipative e¤ects directly into LSWqs which will be modi…ed to ILBqs [4-5]. We put 2 (@@x22+
@2 @y2); 2 @ 2 M @x2 and 2 @ 2 N @x2
into Eqs, respectively. The new set of equations with the above additional expressions describes a PDEs problem that we shall restrict our study to the non-advective term, numerical linear dispersion and weak dissipation terms. The new proposed scheme is given by the following procedure:
@ @t + @M @x + @N @y 2 @2 @x2 + @2 @y2 = o ( t; x; y) ; (34) @M @t + gh @ @x+ gh 2 1 2 @3 @x3+ 2 3 @2 @x @y2 + 3 @2 @t2 @x 2 h @2M @x2 =; o t3; x3; x y2; t2 x ; (35) @N @t + gh @ @y+ gh 2 1 2 @3 @y3 + 2 3 @2 @x2@y + 3 @2 @t2 @y 2 h @2N @y2 =; = o t3; y3; x2 y; t2 y : (36) Dispersion-dissipation scheme
We employ the third-order ADI predictor corrector (PC) scheme [9] for spatial derivatives with three-time level. As a result, both numerical weakly dissipation and linear dispersion are kept with good precision.
The x -sweep equations, in which k+12
i;j and M k+1
2
i+1 2;j
are the unknown variables: Preditor stage k+1 2 i;j ki;j t 2 + Mk+12 i+1 2;j Mk+12 i 1 2;j x + Ni;j+k 1 2 N k i;j 1 2 y ; = 2 ( x)2 k+1 2 i+1;j 2 k+1 2 i;j + k+1 2 i 1;j+ k
i;j+1 2 ki;j+ ki;j 1 : (37)
Corrector Stage Mk+ 1 2 i+1 2;j Mk i+1 2;j t 2 + gh k+1 2 i+1;j k+1 2 i;j x + 1 12 xgh k+1 2 i+2;j 3 k+1 2 i+1;j+ 3 k+1 2 i;j ; + k+12 i 1;j + 2 12 xgh h k+1 2 i+1;j+1 2 k+1 2 i+1;j+ k+1 2 i+1;j 1 k+1 2 i;j+1 2 k+1 2 i;j ; + k+12 i;j 1 i + 3 4 xgh h k+1 i+1;j 2 k+1 2 i+1;j+ k i+1;j k+1i;j 2 k+1 2 i;j + k i;j i ; = 2 h ( x)2 M k+1 2 i+3 2;j 2Mk+12 i+1 2;j + Mk+12 i 1 2;j : (38)
The y-sweep equations, in which k+1i;j and Mk+1
i;j1 2
Predictor stage k+1 i;j k+1 2 i;j t 2 + Mk+ 1 2 i+1 2;j M k+1 2 i 1 2;j x + Ni;j+k+11 2 N k+1 i;j 1 2 y ; = 2 ( x)2 k+1 2 i+1;j 2 k+1 2 i;j + k+1 2 i 1;j+ k+1 i;j+1 2 k+1 i;j + k+1 i;j 1 : (39)
The y-sweep equations, in which k+1i;j and Mi;jk+11 2
are the unknown variables: Corrector stage Ni;j+k+11 2 N k+1 2 i;j+1 2 t 2 + gh k+1 i;j+1 k+1i;j y + 1 12 ygh k+1
i;j+2 3 k+1i;j+1+ 3 k+1i;j ;
+ k+1i;j 1 + 2 12 ygh
k+1
i+1;j+1 2 k+1i;j+1+ k+1i 1;j+1 k+1i+1;j 2 k+1i;j ;
+ k+1i 1;j + 3 4 ygh h k+1 i;j+1 2 k+1 2
i;j+1+ ki;j+1 k+1i;j 2 k+1 2 i;j + ki;j i ; = 2 h ( y)2 N k+1 i;j+3 2 2Ni;j+k+11 2 + Ni;jk+11 2 : (40)
Axand Ay are the discretization matrices.
The rank of the matrix is di¤erent of unknown numbers, then the solution of (41)-(47) system is not unique. For that, we support the work Harter[11] that a general uniqueness proof is now known to be unobtainable.
Convergence, Stability and Consistency
The schemet ADI de…ned by Eqs .(37)-(40) is stable (see, [4, 5]. By obtaining the local discretization error and the well-known classical theorem [9], we con-cluded that,the scheme is convergent.
The solution of the Eqs. (37)-(40) can be written in the following Fourier forms, (see [13]):
= 0 teimxeily; (48)
M = M0 teimxeily;
N = N0 teimxeily
In the stability analysis, the ampli…cation factor, j tj, should be less than or
equal to unity.(see [4, 5])
TThe intermediate ADI solution introduces an added complication, thus we can either combine separate estimates of the local discretization errors of the predictor and corrector steps, [9].
4
Numerical Results and Discussion
The goal of this numerical study is recognition of mathematical aspect when non local terms give a more realistic pro…le to dispersive free surface of tsunami. Sur-pass the obstacle that appears in the experimental when we introduce delicate values of , i.e., values of in nature.There are some time intervals for which the solution is Bichromatic Wave (BW). These intervals intersect only at the points x [0; 15m], so that BW for this mode is established only at these values.
4.1
Free Surface Pro…le
Following this application, our simulation focuses at waves dispersive a¤ected by weak dissipation. The choice of the kinematic viscosity for the practical simulations of water waves is not obvious. However, in various experimental and theoretical studies, researchers independently concluded that, a value of = 10 3m2=s …ts very well available data (see, [10]). The most recent experimental
The relative error (rerr) is de…ned by following relationship: rerr =
P x i=1
P y
j=1[ (i;j) ref(i;j)]2
P x i=1
P y
j=1[ ref(i;j)]2
where, ref(i; j) is calculated at x = 1/1000 as a reference among the pro-posed solutions.
Figure 3 and Table 1, show that the error decreases gradually as x ap-proaches zero. Then, the error between the approximate solution and the ex-act solution decreases corresponding to the theoretical convergence of order O(( x)3; ( x)( t)2; ( x)2( t)), (see section:Convergence of the Scheme).The free surface pro…le for inviscid and weakly viscous ‡uid.
Comparison the free surface pro…le obtained and the numerical scheme of [12]
5
Conclusion and remarks
model, we proposed the introducing an additional term to the novel modi…ed scheme. This latter is comparatively has some limitations, i.e., the grid size has to be equal in both horizontal directions and check stability criteria 33. In other words, our approach calculates the dispersion-correction factors instead of choosing spatial grid size and step size again to mimic the frequency dispersion of the ILBqs, [12].
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