NUMERICAL METHOD FOR A SINGULARLY PERTURBED DIFFERENTIAL SYSTEM

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DIFFERENTIAL SYSTEM

I.G. AMIRALIYEVA

Communicated by the former editorial board

A second-order accurate dierence scheme is developed to solve Boussinesq system. For the time integration, a Crank-Nicolson type scheme is used. The error estimates for the numerical solution are obtained. Numerical results are also presented.

AMS 2010 Subject Classication: 65M06, 65M12, 65M15.

Key words: Boussinesq system, dierence scheme, error estimates.

1. INTRODUCTION

We shall be concerned with developing dierence schemes for approximating the solution {u(x, t), v(x, t)} , arising in the long water waves theory.

L

₁

[u, v] := ∂u

∂t − ∂

³

u

∂t∂x

²

+ αu ∂u

∂x + a

₀

(x, t)u + a

₁

(x, t)v + a

₂

(x, t) ∂v

∂x = f

₁

(x, t),

(1.1) (x, t) ∈ Q,

L

2

[u, v] := ∂v

∂t − ∂

³

v

∂t∂x

²

+ β ∂

∂x (uv) + b

0

(x, t)v +b

1

(x, t)u +b

2

(x, t) ∂u

∂x = f

2

(x, t),

(1.2) (x, t) ∈ Q,

(1.3) u(x, 0) = ϕ(x), v(x, 0) = ψ(x), x ∈ Ω, (1.4) u(0, t) = v(0, t) = u(l, t) = v(l, t) = 0, t ∈ (0, T ],

where Q = Ω ∪ (0, T ], Ω = (0, l) and Ω the closure of Ω. a

k

(x, t), b

k

(x, t)(k = 0, 1, 2), f

_k

(x, t)(k = 1, 2), ϕ(x), ψ(x) are given suciently smooth function such

as ∂

^s

a

k

∂x

^s

, ∂

^s

b

k

∂x

^s

, ∂

^s

a

k

∂t

^s

, ∂

^s

b

k

∂t

^s

∈ C(Q)(k, s = 0, 1, 2)

(1.5) ∂

^s

f

_k

∂x

^s

, ∂

^s

f

_k

∂t

^s

, ∈ C(Q)(k = 1, 2; s = 0, 1, 2)

MATH. REPORTS 15(65), 3 (2013), 243248

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and α , β are given constants.

The system (1.1)(1.2) provides an approximate model for long gravity waves of small amplitude in one dimension [11, 5]. Here, u denotes horizon- tal velocity at the level of uid, and v is the height of the surface above the undisturbed level.

The dierence methods having rst-order in time direction accuracy was investigated in [1, 2]. In [2] two-dimentional case and in [1] singularly perturbed case was considered. Note also, that many studies have been devoted to the so- called Sobolev or pseudo-parabolic equation [3, 4, 610, 12] (see, also references cited in them).

In this paper, we present a dierence method with exponential tting factors to problem (1.1)(1.4). For the time variable we use Crank-Nicolson type discretization. The fully discrete scheme is shown to be second-order both accurate in space and time variables.

2. DIFFERENCE APPROXIMATION AND CONVERGENCE

Let a set of mesh nodes that discretises Q be given by ω

_hτ

= ω

_h

× ω

_τ

with

ω

_h

= {x

_i

= ih, i = 1, 2, ..., N − 1, h = l/N } , ω

τ

= {t

_j

= jτ, i = 1, 2, ..., N

0

, τ = T /N

0

} and

ω

⁺_h

= ω

h

∪ {x

_N

= l}, ω ¯

h

= ω

h

∪ {x

₀

= 0, x

N

= l},

¯

ω

τ

= ω

τ

∪ {t

₀

= 0}, ω ¯

hτ

= ¯ ω

h

× ω ¯

τ

.

Dene the following nite dierences for any mesh function g

_i

= g(x

_i

) given on ω ¯

_h

by

g

x,i¯

= g

i

− g

i−1

h , g

x,i

= g

i+1

− g

i

h , g

0

x,i

= g

i+1

− g

i−1

2h , g

xx,i¯

= g

i+1

− 2g

i

+ g

i−1

h

²

.

Introduce the inner products for the mesh functions v

i

and w

i

dened on

¯

ω

_h

as follows

(v, w)

0

≡ (v, w)

ω_h

:=

N−1

X

i=1

hv

i

w

i

,

(v, w] ≡ (v, w)

_ω+ h

:=

N

X

i=1

hv

_i

w

_i

.

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For any mesh function v

_i

, vanishing for i = 0 and i = N we introduce the norms

kvk

²₀

≡ kvk

²_0,ω

h

:= (v, v)

0

, kvk

²0

1

:= (v

x¯

, v

x¯

], kvk

²₁

: = kvk

²₀

+ kvk

²₀

1

,

Given a function g ≡ g

_i^j

≡ g(x

_i

, t

_j

) dened on ω ¯

_hτ

, we shall also use the notation

g

_¯_t,i^j

= g

^j_i

− g

^j−1_i

τ , g

^j_t,i

= g

_i^j+1

− g

_i^j

τ .

On the mesh ω

_hτ

, we approximate (1.1)(1.4) by the following dierence problem of Crank-Nicolson type

l

1

[y

1

, y

2

] := y

_1¯_t

−θy

_1¯_t¯_xx

+αS

^(0.5)

(y

1

)+a

^(0.5)₀

y

₁^(0.5)

+a

^(0.5)₁

y

^(0.5)₂

+a

^(0.5)₂

y

2x⁰

= f

₁^(0.5)

,

(2.1) (x, t) ∈ ω

_hτ

,

l

₂

[y

₁

, y

₂

] := y

_2¯_t

− θy

_2¯_t¯_xx

+ β (y

₁

y

₂

)

0

x

+ b

^(0.5)₀

y

^(0.5)₂

+ b

^(0.5)₁

y

₁^(0.5)

+ b

^(0.5)₂

y

1x⁰

= f

₂^(0.5)

,

(2.2) (x, t) ∈ ω

_hτ

,

(2.3) y

₁

(x, 0) = ϕ(x), y

₂

(x, 0) = ψ(x), x ∈ ω ¯

_h

, (2.4) y

1

= y

2

= 0, for x = 0, ` and t ∈ ω ¯

τ

, where

g

^(0.5)

≡ g

^(0.5)j_i

= 1 2

g

_i^j

+ g

_i^j−1

, S(y

^j_1,i

) = 1

3 y

₁²

j

x,i0

+ y

^j_1,i

y

^j

1x,i⁰

, θ = h

²

4sinh

²

(h/2) .

Our method of approximating the convective term in equation (1.1) has the property that (S(y

1

), y

1

)

ωh

= 0 , i.e., it preserves the analogous principle for the dierential case.

We note that assumption (1.5) implies the existence of a unique solution {u, v} to problem (1.1)(1.4), belonging to C

³

( ¯ Q) . Then, by the similar manner to that in [14 ], it can be shown that

`

_k

[u, v] + R

_k

= f

_k

, k = 1, 2; (x, t) ∈ ω

_hτ

with truncation errors R

k

, such that

kR

_k

k

₀

= O(h

²

+ τ

²

), k = 1, 2; t ∈ ω

_τ

.

Further, from the identity

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τ X

p=1

(`

1

[y

₁^p

, y

₂^p

] − `

1

[u

^p

, v

^p

], z

₁^p

)

₀

+ τ X

p=1

(`

2

[y

^p₁

, y

^p₂

] − `

2

[u

^p

, v

^p

], z

^p₂

)

₀

= τ

j

X

p=1 2

X

k=1

R

_k

, z

_k^p

0

,

using summation by parts and appropriate dierence embedding inequalities, for the errors z

₁

= y

₁

− u, z

₂

= y

₂

− v by the similar technique described in [1, 2], we have

(2.5) δ

j

≤ C





 δ

∗

+ τ

j

X

p=1

δ

j

+ δ

j−1

+ δ

_j²

+ δ

_j−1²





 with

δ

j

=

2

X

k=1

θ

z

^j_k

2

0

1

+ kz

_k

k

²₀

(δ

0

= 0),

δ

∗

= τ X

wτ

2

X

k=1

kR

_k

k

²₀

and constant C, that is independent of the mesh parameters. From (2.5), by an application of the dierence analogue of Bernoulli inequality (see, also [1, 2]), for suciently small τ , we obtain

δ

_j

≤ Cδ

∗

, j = 1, 2, ..., N

₀

. We summarize this result in the following theorem.

Theorem 2.1. Under the conditions on a

_k

, b

_k

, f

_k

indicated in (1.5), the solution of (2.1)(2.4) converges to the exact solution in the mesh norm k.k

₁

with the rate O(h

²

+ τ

²

) :

ky

₁

− uk

₁

+ ky

₂

− vk

₁

≤ C(h

²

+ τ

²

).

3. NUMERICAL RESULTS

In this section, we present numerical results obtained by applying the numerical method (2.1)(2.4) to a particular problem

∂u

∂t − ∂

³

u

∂t∂

²

x + 3u ∂u

∂x + (e

^−t

+ x

²

)u = f (x, t), (x, t) ∈ [0, 1] × [0, 1], u(x, 0) = sin(πx) − 0.5x

²

(1 − x),

u(0, t) = u(1, t) = 0,

where f is such that the exact solution is given by u(x, t) = − 1

2 x

²

(1 − x) + e

^−t

sin πx.

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In the computations, the quasilinearization technique to the dierence equation was used (see, [3]). The step sizes h and τ were both taken to be 0.025. The results at selected points are presented in Table 1.

Table 1

(x, t) Exact Solution Approximate Solution Pointwise Errors (0.05,0.05) 0.14761757 0.14761772 0.00000015 (0.15,0.15) 0.38119074 0.38119121 0.00000047 (0.25,0.25) 0.52725781 0.52725907 0.00000126 (0.35,0.35) 0.58806919 0.58807119 0.00000200 (0.45,0.45) 0.57409039 0.57409267 0.00000228 (0.55,0.55) 0.50178410 0.50178641 0.00000231 (0.65,0.65) 0.39120869 0.39121092 0.00000223 (0.75,0.75) 0.26370109 0.26370237 0.00000128 (0.85,0.85) 0.13985482 0.13985580 0.00000098 (0.95,0.95) 0.03793713 0.03793732 0.00000019

The obtained results show an excellent degree of accuracy for the above described method.

Acknowledgment. This study is partially supported by Universiti Brunei Darussalam under Grant UBD/PNC2/2/RG/1(159).

REFERENCES

[1] G.M. Amiraliyev, On the numerical solution of the system of Boussinesque with boundary layers. USSR Modelling in Mechanics 3 (1988), 314.

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[8] W.H. Ford and T.W. Ting, Uniform error estimates for dierence approximations to nonlinear pseudo-parabolic partial dierential equations. SIAM J. Numer. Anal. 15 (1974), 155169.

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[9] M. Yang, Analysis of second order nite volume element methods for pseudo-parabolic equations in three spatial dimensions. Appl. Math. Comput. 196 (2008), 94104.

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Received 16 September 2011 Sinop University,

Faculty of Sciences, Department of Statistics,

5700 Sinop, Turkey ailhame@gmail.com