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On the convergence of difference schemes for the Benjamin–Bona–Mahony (BBM) equation
Applied Mathematics and Computation 182 (2006) 999–1005 www.elsevier.com/locate/amc
On the convergence of difference schemes for the Benjamin–Bona–Mahony (BBM) equation T. Achouri a, N. Khiari a, K. Omrani
b,*
a
b
Faculte´ des Sciences de Monastir, route de Kairouan, 5000 Monastir, Tunisia Institut Supe´rieur des Sciences, Applique´es et de Technologie de Sousse, 4003 Sousse Ibn Khaldoun, Tunisia
Abstract In this paper, we analyze a Crank–Nicolson-type finite difference scheme for the BBM equation. We prove the secondorder convergence in discrete H1-norm of the difference approximate solutions. The existence, stability and uniqueness are also discussed in detail. At last a linearized difference scheme is given and its convergence is also proved. Ó 2006 Elsevier Inc. All rights reserved. Keywords: BBM equation; Finite difference scheme; Existence; Stability; Uniqueness; Convergence; Linearization
1. Introduction The main purpose of this paper is to study a finite difference discretization of the BBM equation: ut uxxt þ ux þ uux ¼ 0; uð0; tÞ ¼ uð1; tÞ ¼ 0; uðx; 0Þ ¼ u0 ðxÞ;
0 < x < 1; 0 < t 6 T ;
0 < t 6 T;
0 6 x 6 1;
ð1:1aÞ ð1:1bÞ ð1:1cÞ
that has been studied extensively by Benjamin et al. [3] and others [4,10–12,18], as an alternative to Korteweg– de Vries (KdV) equation ut þ uxxx þ ux þ uux ¼ 0;
0 < x < 1; 0 < t 6 T ;
ð1:2Þ
as a model for unidirectional, long, dispersive waves. Several numerical methods have been proposed in the literature for discretizing the problem (1.1), see [21,13,6,19,7,2].
*
Corresponding author. E-mail address: [email protected] (K. Omrani).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.04.069
1000
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
1 Let N, J be any positive integers and h ¼ J þ1 ; k ¼ NT ; xj ¼ jh; j = 0, 1, . . . , J + 1 and tn = nk,
1 tnþ2
¼ ðn þ 12Þk; unj ¼ uðxj ; tn Þ; un ¼ ðun0 ; . . . ; unJ þ1 Þ for n = 0, . . . , N. Let RJ0 þ2 ¼ fv ¼ ðv0 ; . . . ; vJ þ1 Þ 2 RJ þ2 ;
v0 ¼ vJ þ1 ¼ 0g:
We define the difference operators as, for a function v 2 RJ0 þ2 vj vj1 vjþ1 vj ; rþ ; r h vj ¼ h vj ¼ h h vjþ1 vj1 vjþ1 2vj þ vj1 ; Dh vj ¼ rh vj ¼ ; 2h h2 vnþ1 vnj vnþ1 þ vnj nþ1 j j ; vj 2 ¼ : ot vnj ¼ k 2 For v; w 2 RJ0 þ2 , we define a discrete inner product as ðv; wÞh ¼ h
J X
vj wj :
j¼1
The discrete L2-norm k Æ kh, H1-semi-norm | Æ |1,h and H1-norm k Æ k1,h are defined, respectively, as " #12 J X kvkh ¼ h v2j ; j¼1
"
jvj1;h
#12 J X 2 þ ¼ h ðrh vj Þ ; j¼0
h
i12 2 2 kvk1;h ¼ kvkh þ jvj1;h : For a simple notation, let u be a function defined as uðv; wÞj ¼
1 ðvj1 þ vj þ vjþ1 Þðwjþ1 wj1 Þ: 6h
Then we may easily seen that the following relations hold as in [1,8,9], for v; w 2 RJ0 þ2 : ðv; rh wÞh ¼ ðrh v; wÞh ; J X þ ðrþ ðDh v; wÞh ¼ h h vj Þðrh wj Þ;
ð1:3Þ ð1:4Þ
j¼1
ðDh v; vÞh ¼ jvj21;h ;
ð1:5Þ
ðrh v; vÞh ¼ 0;
ð1:6Þ
ðuðv; vÞ; vÞh ¼ 0:
ð1:7Þ
Using the above notations, we discretize problem (1.1a)–(1.1c) by the following Crank–Nicolson-type finite difference scheme, we define approximations U n 2 RJ0 þ2 of un recursively by: nþ1 nþ1 nþ1 ot U nj Dh ðot U nj Þ þ rh U j 2 þ u U j 2 ; U j 2 ¼ 0; 1 6 j 6 J ; 0 6 n 6 N 1; ð1:8aÞ U 0j ¼ u0 ðxj Þ;
1 6 j 6 J:
ð1:8bÞ
In this article, we will prove that scheme (1.8) is convergent at the rate of (h2 + k2) in the discrete H1-norm. The existence and stability of the difference solution (1.8) is shown in Section 2. The error in H1-norm between the exact solution and the approximation solution is studied in Section 3. Uniqueness of the difference solution is presented in Section 4. At last section, a second-order convergent extrapolated method is proved.
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
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2. Existence and stability 2.1. Existence In this section, we shall use the following Brower fixed-point theorem [1,5] in order to show the existence of solutions of the Crank–Nicolson difference scheme (1.8). Theorem 1. Let (H, (Æ, Æ)h) be a finite dimensional inner product space, k Æ kh the associated norm, and g : H ! H be continuous. Assume moreover that 9a > 0
8x 2 H ;
kxkh ¼ a;
ðgðxÞ; xÞh P 0:
Then, there exists a x* 2 H such that g(x*) = 0 and kx*kh 6 a. Theorem 2. The solution Un of the Crank–Nicolson finite difference scheme (1.8a) and (1.8b) exist. Proof. The argument of existence of the Crank–Nicolson approximations proceeds in an inductive way. For fixed n, we rewrite (1.8) in the form nþ1 nþ1 nþ1 nþ1 nþ1 2 U j 2 U nj 2Dh U j 2 U nj þ krh U j 2 þ ku U j 2 ; U j 2 ¼ 0; 1 6 j 6 J : ð2:1Þ The mapping g : RJ0 þ2 ! RJ0 þ2 ðgðvÞÞj ¼ 2vj 2U nj 2Dh vj þ 2Dh U nj þ krh vj þ kuðvj ; vj Þ;
ð2:2Þ
16j6J
is obviously continuous. Taking in (2.2) the inner product with v, we obtain from (1.4) and (1.5) 2
2
ðgðvÞ; vÞh ¼ 2kvkh 2ðU n ; vÞh þ 2jvj1;h 2h
J X
þ n ðrþ h U j Þðrh vj Þ þ kðrh v; vÞh þ kðuðv; vÞ; vÞh :
j¼1
It follows from (1.6), (1.7) that: 2
2
ðgðvÞ; vÞh P 2kvkh 2kU n kh kvkh þ 2jvj1;h 2jU n j1;h jvj1;h P 2kvk2h kU n k2h þ kvk2h þ 2jvj21;h jU n j21;h þ jvj21;h ¼ kvk21;h kU n k21;h : 2
1
2
Thus for kvk1;h ¼ kU n k1;h þ 1, there exists (g(v), v)h > 0 . The existence of U nþ2 follows from Theorem 1 and consequently the existence of Un+1 is obtained. h 2.2. Stability We will prove that the finite difference scheme (1.8) is conservative, which implies the stability of the approximate solution. Theorem 3. The solution of (1.8) satisfies the following conservative property: kU n k1;h ¼ kU 0 k1;h : 1
Proof. Taking in (1.8a) the inner product with U nþ2 , using (1.6) and (1.7), we obtain 1 1 2 2 ot kU n kh þ ot jU n j1;h ¼ 0: 2 2 Summing (2.3) from n = 0 to m, we get 2
2
kU mþ1 k1;h ¼ kU 0 k1;h : This completes the proof.
h
ð2:3Þ
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T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
3. Convergence In this section, we consider the convergence of approximate solutions of (1.8a) and (1.8b) to the solutions of (1.1a)–(1.1c). Theorem 4. Let the solution u of (1.1) be sufficiently smooth, and U0, U1, . . . , UN satisfy (1.8). Then, for k sufficiently small, kun U n k1;h 6 Cðh2 þ k 2 Þ;
ð3:1Þ
where C is a constant independent of h and k. Proof. Making use the Taylor expansion, we find 1 nþ1 nþ nþ1 ot unj Dh ot unj þ rh uj 2 þ u uj 2 ; uj 2 ¼ rnj ; u0j ¼ u0 ðxj Þ;
1 6 j 6 J ; 0 6 n 6 N 1;
ð3:2aÞ ð3:2bÞ
1 6 j 6 J;
where rn 2 RJ0 þ2 is the truncation errors of the difference scheme (1.8) and there exists a positive constant C0 such that jrnj j 6 C 0 ðh2 þ k 2 Þ:
ð3:3Þ
Let enj ¼ unj U nj ; 0 6 n 6 N ; 1 6 j 6 J : Subtracting (1.8) from (3.2), we obtain 1 nþ1 nþ1 nþ1 nþ nþ1 ot enj Dh ot enj þ rh ej 2 ¼ u U j 2 ; U j 2 u uj 2 ; uj 2 þ rnj ; 1 6 j 6 J ; e0j ¼ 0;
ð3:4aÞ ð3:4bÞ
1 6 j 6 J: 1
Taking in (3.4a) the inner product with enþ2 , we obtain 1 1 1 1 1 1 1 1 1 1 1 ot ken k2h þ ot jen j21;h þ ðrh enþ2 ; enþ2 Þh ¼ ðuðU nþ2 ; U nþ2 Þ; enþ2 Þh ðuðunþ2 ; unþ2 Þ; enþ2 Þh þ ðrn ; enþ2 Þh : 2 2 Noting that from (1.7) 1 1 1 1 1 1 u U nþ2 ; U nþ2 ; enþ2 u unþ2 ; unþ2 ; enþ2 h h 1 1 1 1 1 1 1 1 1 ¼ u enþ2 ; enþ2 ; enþ2 u enþ2 ; unþ2 ; enþ2 u unþ2 ; enþ2 ; enþ2 h h h h i 1 1 1 6 C jenþ2 j1;h þ kenþ2 kh kenþ2 kh 1 2 1 2 6 C jenþ2 j1;h þ kenþ2 kh : Using the above inequality (3.5), (1.6) and (3.3) h i 1 1 ot ken k2h þ ot jen j21;h 6 C jen j21;h þ jenþ1 j21;h þ ken k2h þ kenþ1 k2h þ ðh2 þ k 2 Þ2 : 2 2 Then, (3.1) follows is view of Gronwall’s discrete inequality and (3.4b). h 4. Uniqueness We shall show uniqueness of the approximations U1, U2, . . . , Un satisfying (1.8). Theorem 5. The solution Un of the finite difference scheme (1.8) is unique.
ð3:5Þ
ð3:6Þ
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
1003
Proof. Suppose that V0 = u0 and let V n 2 RJ0 þ2 be another solution of (1.8). Then Vn satisfies 1 1 1 ot V n Dh ðot V n Þ þ rh V nþ2 þ u V nþ2 ; V nþ2 ¼ 0:
ð4:1Þ
Let hn = Vn Un with h0 = 0, we obtain from (1.8a) and (4.1) 1 1 1 1 1 ot hn Dh ðot hn Þ þ rh hnþ2 ¼ u U nþ2 ; U nþ2 u V nþ2 ; V nþ2 :
ð4:2Þ
The proof of uniqueness is with the mathematical induction. Now supposing hn = 0 and taking in (4.2) the 1 inner product with hnþ2 , rewriting the terms on the right-hand side according to (3.6), using (1.6) and (1.7) and applying the Schwarz inequality, we obtain h i 2 2 2 2 2 2 ot khn kh þ ot jhn j1;h 6 C jhn j1;h þ jhnþ1 j1;h þ khn kh þ khnþ1 kh : Since h0 = 0, it follows from the discrete Gronwall inequality that for k0 such that 1 Ck > 0 for 0 < k < k0: khnþ1 k1;h ¼ 0: This completes the proof of the uniqueness of solutions for (1.8).
h
5. A linearized difference scheme We construct for (1.8) the following linearized three level difference scheme (see [14–17]): nþ1 ot U nj Dh ot U nj þ rh U j 2 þ u U ^nj ; U ^nj ¼ 0; 1 6 j 6 J ; 1 6 n 6 N 1; U 0j ¼ u0 ðxj Þ; 1 6 j 6 J ; d2 u0 du0 1 1 ðxj Þ þ uðu0 ðxj Þ; u0 ðxj ÞÞ ; U j Dh U j ¼ u0 ðxj Þ 2 ðxj Þ k dx dx
ð5:1aÞ ð5:1bÞ
1 6 j 6 J;
ð5:1cÞ
where U ^n ¼ 32 U n 12 U n1 ; 1 6 n 6 N : For a smooth function u, we have 3 1 1 u^n ¼ un un1 ¼ un2 þ Oðk 2 Þ 2 2
as k ! 0:
ð5:2Þ
Next we need the following Lemma (Sun [20]). Lemma 1. Let b1,b2 and ai, i = 1, 2, 3, . . ., be positive and satisfy aiþ1 6 ð1 þ b1 kÞai þ b2 k; then aiþ1
i ¼ 1; 2; 3; . . . ;
b2 6 expðb1 ikÞ a1 þ ; b1
i ¼ 1; 2; 3; . . .
Theorem 6. Suppose that the solution u(x, t) of (1.1) is sufficiently regular. Then, for k sufficiently small, the solution of linearized modification (5.1) is convergent to the solution of problem (1.1) and the order of convergence rate is O(h2 + k2) in the H1-discrete-norm. Proof. From Taylor expansion, we have nþ1 ot unj Dh ot unj þ rh uj 2 þ u u^nj ; u^nj ¼ P nj ; u0j ¼ u0 ðxj Þ; 2
u1j Dh ðu1j Þ ¼ u0 ðxj Þ
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:3bÞ
d u0 du0 ðxj Þ þ uðu0 ðxj Þ; u0 ðxj ÞÞ þ Qj ; ðxj Þ k 2 dx dx
ð5:3aÞ
1 6 j 6 J;
ð5:3cÞ
1004
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
where P nj and Qj are the truncation errors of difference scheme (5.1). Using (5.2), there exist a positive constant c0 independent of step sizes such that jP nj j 6 c0 ðh2 þ k 2 Þ;
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:4aÞ
jQj j 6 c0 ðh2 þ k 2 Þ;
1 6 j 6 J:
ð5:4bÞ
Let Enj ¼ unj U nj , we obtain from (5.1) and (5.3) nþ1 ot Enj Dh ot Enj þ rh Ej 2 ¼ u U ^nj ; U ^nj u u^nj ; u^nj þ P nj ; E0j ¼ 0; E1j
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:5bÞ
1 6 j 6 J;
¼ Qj ;
ð5:5aÞ
ð5:5cÞ
1 6 j 6 J:
We will prove by inductive method that kEn k1;h 6 Cðu; T Þðh2 þ k 2 Þ;
ð5:6Þ
0 6 n 6 N:
From (5.5b), we have kE0 k1;h ¼ 0:
ð5:7Þ
Using (5.4b) and (5.5c), we obtain kE1 kh 6 c0 ðh2 þ k 2 Þ:
ð5:8Þ
Now from (5.5c) and Taylor expansion with integration term and by means of (1.1), we get Z 1 sutt ðxj ; k skÞ ds Qj ¼ k 2 0
and rþ h Qj ¼
1 Qjþ1 Qj ¼ k 2 h
Z 0
1
s
utt ðxjþ1 ; k skÞ utt ðxj ; k skÞ ds ¼ k 2 h
Z
1
suttx ðnj ; k skÞ ds;
0
where nj 2 (xj,xj+1). Therefore jE1 j1;h 6 Cðh2 þ k 2 Þ:
ð5:9Þ
Using (5.8) and (5.9) kE1 k1;h 6 Cðh2 þ k 2 Þ:
ð5:10Þ
It follows from (5.7) and (5.10) that (5.6) is valid for n = 0 and n = 1. Now suppose that (5.6) is true for 1 1 6 n 6 l, and taking in (5.5a) the inner product with Enþ2 , we obtain from (1.6) and the boundedness of ux with along line of the proof of Theorem 4 h i 1 2 2 2 2 2 2 ot kEn kh þ ot jEn j1;h 6 C kEn1 k1;h þ kEn k1;h þ kEnþ1 k1;h þ kP n kh : 2 This yields by (5.4a) h i 1 nþ1 2 kE k1;h kEn k21;h 6 C kEn1 k21;h þ kEn k21;h þ kEnþ1 k21;h þ c20 ðh2 þ k 2 Þ2 2k or 2
2
2
2
ð1 2CkÞkEnþ1 k1;h 6 ð1 þ 2CkÞkEn k1;h þ 2CkkEn1 k1;h þ 2c20 kðh2 þ k 2 Þ : 1 Whence, for k 6 6C
kEnþ1 k21;h 6 ð1 þ 6CkÞkEn k21;h þ 3CkkEn1 k21;h þ 3c20 kðh2 þ k 2 Þ2 :
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
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Here, by above 2 2 2 2 2 max kEnþ1 k1;h ; kEn k1;h 6 ð1 þ 9CkÞ max kEn k1;h ; kEn1 k1;h þ 3c20 kðh2 þ k 2 Þ : By Lemma 1, we have c2 2 2 2 2 2 max kElþ1 k1;h ; kEl k1;h 6 expð9CklÞ maxðkE1 k1;h ; kE0 k1;h Þ þ 0 ðh2 þ k 2 Þ : 3C Using (5.7) and (5.10), we find c2 2 2 2 max kElþ1 k1;h ; kEl k1;h 6 expð9CT Þ C 2 þ 0 ðh2 þ k 2 Þ : 3C This yields kElþ1 k1;h 6 Cðu; T Þðh2 þ k 2 Þ: That means, (5.6) is valid for n = l + 1. This completes the proof.
h
References [1] G.D. Akrivis, Finite difference discretization of the Kuramoto–Sivashinsky equation, Numer. Math. 63 (1992) 1–11. [2] D.N. Arnold, J. Douglas Jr., V. Thome´e, Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comput. 27 (1981) 737–743. [3] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. Lond. A 272 (1972) 47–78. [4] J.L. Bona, R. Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. Lond. A 278 (1975) 555– 601. [5] F.E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, in: Applications of Nonlinear Partial Differential Equations, in: R. Finn (Ed.), Proc. Sympos. Appl. Math., vol. 17, Amer. Math. Soc., Providence, 1965, pp. 24–49. [6] J.C. Eilbeck, G.R. McGuire, Numerical studies of the regularized long-wave equations I: numerical methods, J. Comput. Phys. 19 (1975) 43–57. [7] R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, SIAM J. Numer. Anal. 15 (1978) 1125–1150. [8] R. Kannan, S.K. Chung, Finite difference approximate solutions for the two-dimensional Burger’s system, Comput. Math. Applicat. 44 (2002) 193–200. [9] J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal. 27 (1990) 20–31. [10] L.A. Medeiros, M. Milla Miranda, Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl. 59 (1977) 432–441. [11] L.A. Medeiros, G. Perla Menzela, Existence and uniqueness for periodic solutions of the Benjamin–Bona–Mahony equation, SIAM J. Math. Anal 8 (1977) 792–799. [12] M. Milla Miranda, Weak solutions of a modified KdV equation, Bol. Soc. Brasil. Mat. 6 (1975) 57–63. [13] K. Omrani, The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.12.046. [14] K. Omrani, On fully discrete Galerkin approximations for the Cahn–Hilliard equation, Math. Model. Anal. 9 (4) (2004) 313–326. [15] K. Omrani, Convergence of Galerkin Approximations for the Kuramoto–Tsuzuki equation, Numer. Method Partial Differ. Equ. 21 (5) (2005) 961–975. [16] K. Omrani, A second order accurate difference scheme on nonuniform meshes for nonlinear heat-conduction equation, Far East J. Appl. Math. 20 (3) (2005) 355–365. [17] K. Omrani, M.L. Ben Mohamed, A linearized difference scheme for the Sivashinsky equation, Far East J. Appl. Math. 20 (2) (2005) 179–188. [18] D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966) 321–330. [19] M.A. Raupp, Galerkin methods applied to the Benjamin–Bona–Mahony equation, Bol. Soc. Brasil. Mat. 6 (1975) 65–77. [20] Z.Z. Sun, A second-order accurate linearized difference scheme for the two-dimensional Cahn–Hilliard equation, Math. Comput. 64 (212) (1995) 1463–1471. [21] L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975) 289–303.
On the convergence of difference schemes for the Benjamin–Bona–Mahony (BBM) equation T. Achouri a, N. Khiari a, K. Omrani
b,*
a
b
Faculte´ des Sciences de Monastir, route de Kairouan, 5000 Monastir, Tunisia Institut Supe´rieur des Sciences, Applique´es et de Technologie de Sousse, 4003 Sousse Ibn Khaldoun, Tunisia
Abstract In this paper, we analyze a Crank–Nicolson-type finite difference scheme for the BBM equation. We prove the secondorder convergence in discrete H1-norm of the difference approximate solutions. The existence, stability and uniqueness are also discussed in detail. At last a linearized difference scheme is given and its convergence is also proved. Ó 2006 Elsevier Inc. All rights reserved. Keywords: BBM equation; Finite difference scheme; Existence; Stability; Uniqueness; Convergence; Linearization
1. Introduction The main purpose of this paper is to study a finite difference discretization of the BBM equation: ut uxxt þ ux þ uux ¼ 0; uð0; tÞ ¼ uð1; tÞ ¼ 0; uðx; 0Þ ¼ u0 ðxÞ;
0 < x < 1; 0 < t 6 T ;
0 < t 6 T;
0 6 x 6 1;
ð1:1aÞ ð1:1bÞ ð1:1cÞ
that has been studied extensively by Benjamin et al. [3] and others [4,10–12,18], as an alternative to Korteweg– de Vries (KdV) equation ut þ uxxx þ ux þ uux ¼ 0;
0 < x < 1; 0 < t 6 T ;
ð1:2Þ
as a model for unidirectional, long, dispersive waves. Several numerical methods have been proposed in the literature for discretizing the problem (1.1), see [21,13,6,19,7,2].
*
Corresponding author. E-mail address: [email protected] (K. Omrani).
0096-3003/$ - see front matter Ó 2006 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2006.04.069
1000
T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
1 Let N, J be any positive integers and h ¼ J þ1 ; k ¼ NT ; xj ¼ jh; j = 0, 1, . . . , J + 1 and tn = nk,
1 tnþ2
¼ ðn þ 12Þk; unj ¼ uðxj ; tn Þ; un ¼ ðun0 ; . . . ; unJ þ1 Þ for n = 0, . . . , N. Let RJ0 þ2 ¼ fv ¼ ðv0 ; . . . ; vJ þ1 Þ 2 RJ þ2 ;
v0 ¼ vJ þ1 ¼ 0g:
We define the difference operators as, for a function v 2 RJ0 þ2 vj vj1 vjþ1 vj ; rþ ; r h vj ¼ h vj ¼ h h vjþ1 vj1 vjþ1 2vj þ vj1 ; Dh vj ¼ rh vj ¼ ; 2h h2 vnþ1 vnj vnþ1 þ vnj nþ1 j j ; vj 2 ¼ : ot vnj ¼ k 2 For v; w 2 RJ0 þ2 , we define a discrete inner product as ðv; wÞh ¼ h
J X
vj wj :
j¼1
The discrete L2-norm k Æ kh, H1-semi-norm | Æ |1,h and H1-norm k Æ k1,h are defined, respectively, as " #12 J X kvkh ¼ h v2j ; j¼1
"
jvj1;h
#12 J X 2 þ ¼ h ðrh vj Þ ; j¼0
h
i12 2 2 kvk1;h ¼ kvkh þ jvj1;h : For a simple notation, let u be a function defined as uðv; wÞj ¼
1 ðvj1 þ vj þ vjþ1 Þðwjþ1 wj1 Þ: 6h
Then we may easily seen that the following relations hold as in [1,8,9], for v; w 2 RJ0 þ2 : ðv; rh wÞh ¼ ðrh v; wÞh ; J X þ ðrþ ðDh v; wÞh ¼ h h vj Þðrh wj Þ;
ð1:3Þ ð1:4Þ
j¼1
ðDh v; vÞh ¼ jvj21;h ;
ð1:5Þ
ðrh v; vÞh ¼ 0;
ð1:6Þ
ðuðv; vÞ; vÞh ¼ 0:
ð1:7Þ
Using the above notations, we discretize problem (1.1a)–(1.1c) by the following Crank–Nicolson-type finite difference scheme, we define approximations U n 2 RJ0 þ2 of un recursively by: nþ1 nþ1 nþ1 ot U nj Dh ðot U nj Þ þ rh U j 2 þ u U j 2 ; U j 2 ¼ 0; 1 6 j 6 J ; 0 6 n 6 N 1; ð1:8aÞ U 0j ¼ u0 ðxj Þ;
1 6 j 6 J:
ð1:8bÞ
In this article, we will prove that scheme (1.8) is convergent at the rate of (h2 + k2) in the discrete H1-norm. The existence and stability of the difference solution (1.8) is shown in Section 2. The error in H1-norm between the exact solution and the approximation solution is studied in Section 3. Uniqueness of the difference solution is presented in Section 4. At last section, a second-order convergent extrapolated method is proved.
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2. Existence and stability 2.1. Existence In this section, we shall use the following Brower fixed-point theorem [1,5] in order to show the existence of solutions of the Crank–Nicolson difference scheme (1.8). Theorem 1. Let (H, (Æ, Æ)h) be a finite dimensional inner product space, k Æ kh the associated norm, and g : H ! H be continuous. Assume moreover that 9a > 0
8x 2 H ;
kxkh ¼ a;
ðgðxÞ; xÞh P 0:
Then, there exists a x* 2 H such that g(x*) = 0 and kx*kh 6 a. Theorem 2. The solution Un of the Crank–Nicolson finite difference scheme (1.8a) and (1.8b) exist. Proof. The argument of existence of the Crank–Nicolson approximations proceeds in an inductive way. For fixed n, we rewrite (1.8) in the form nþ1 nþ1 nþ1 nþ1 nþ1 2 U j 2 U nj 2Dh U j 2 U nj þ krh U j 2 þ ku U j 2 ; U j 2 ¼ 0; 1 6 j 6 J : ð2:1Þ The mapping g : RJ0 þ2 ! RJ0 þ2 ðgðvÞÞj ¼ 2vj 2U nj 2Dh vj þ 2Dh U nj þ krh vj þ kuðvj ; vj Þ;
ð2:2Þ
16j6J
is obviously continuous. Taking in (2.2) the inner product with v, we obtain from (1.4) and (1.5) 2
2
ðgðvÞ; vÞh ¼ 2kvkh 2ðU n ; vÞh þ 2jvj1;h 2h
J X
þ n ðrþ h U j Þðrh vj Þ þ kðrh v; vÞh þ kðuðv; vÞ; vÞh :
j¼1
It follows from (1.6), (1.7) that: 2
2
ðgðvÞ; vÞh P 2kvkh 2kU n kh kvkh þ 2jvj1;h 2jU n j1;h jvj1;h P 2kvk2h kU n k2h þ kvk2h þ 2jvj21;h jU n j21;h þ jvj21;h ¼ kvk21;h kU n k21;h : 2
1
2
Thus for kvk1;h ¼ kU n k1;h þ 1, there exists (g(v), v)h > 0 . The existence of U nþ2 follows from Theorem 1 and consequently the existence of Un+1 is obtained. h 2.2. Stability We will prove that the finite difference scheme (1.8) is conservative, which implies the stability of the approximate solution. Theorem 3. The solution of (1.8) satisfies the following conservative property: kU n k1;h ¼ kU 0 k1;h : 1
Proof. Taking in (1.8a) the inner product with U nþ2 , using (1.6) and (1.7), we obtain 1 1 2 2 ot kU n kh þ ot jU n j1;h ¼ 0: 2 2 Summing (2.3) from n = 0 to m, we get 2
2
kU mþ1 k1;h ¼ kU 0 k1;h : This completes the proof.
h
ð2:3Þ
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3. Convergence In this section, we consider the convergence of approximate solutions of (1.8a) and (1.8b) to the solutions of (1.1a)–(1.1c). Theorem 4. Let the solution u of (1.1) be sufficiently smooth, and U0, U1, . . . , UN satisfy (1.8). Then, for k sufficiently small, kun U n k1;h 6 Cðh2 þ k 2 Þ;
ð3:1Þ
where C is a constant independent of h and k. Proof. Making use the Taylor expansion, we find 1 nþ1 nþ nþ1 ot unj Dh ot unj þ rh uj 2 þ u uj 2 ; uj 2 ¼ rnj ; u0j ¼ u0 ðxj Þ;
1 6 j 6 J ; 0 6 n 6 N 1;
ð3:2aÞ ð3:2bÞ
1 6 j 6 J;
where rn 2 RJ0 þ2 is the truncation errors of the difference scheme (1.8) and there exists a positive constant C0 such that jrnj j 6 C 0 ðh2 þ k 2 Þ:
ð3:3Þ
Let enj ¼ unj U nj ; 0 6 n 6 N ; 1 6 j 6 J : Subtracting (1.8) from (3.2), we obtain 1 nþ1 nþ1 nþ1 nþ nþ1 ot enj Dh ot enj þ rh ej 2 ¼ u U j 2 ; U j 2 u uj 2 ; uj 2 þ rnj ; 1 6 j 6 J ; e0j ¼ 0;
ð3:4aÞ ð3:4bÞ
1 6 j 6 J: 1
Taking in (3.4a) the inner product with enþ2 , we obtain 1 1 1 1 1 1 1 1 1 1 1 ot ken k2h þ ot jen j21;h þ ðrh enþ2 ; enþ2 Þh ¼ ðuðU nþ2 ; U nþ2 Þ; enþ2 Þh ðuðunþ2 ; unþ2 Þ; enþ2 Þh þ ðrn ; enþ2 Þh : 2 2 Noting that from (1.7) 1 1 1 1 1 1 u U nþ2 ; U nþ2 ; enþ2 u unþ2 ; unþ2 ; enþ2 h h 1 1 1 1 1 1 1 1 1 ¼ u enþ2 ; enþ2 ; enþ2 u enþ2 ; unþ2 ; enþ2 u unþ2 ; enþ2 ; enþ2 h h h h i 1 1 1 6 C jenþ2 j1;h þ kenþ2 kh kenþ2 kh 1 2 1 2 6 C jenþ2 j1;h þ kenþ2 kh : Using the above inequality (3.5), (1.6) and (3.3) h i 1 1 ot ken k2h þ ot jen j21;h 6 C jen j21;h þ jenþ1 j21;h þ ken k2h þ kenþ1 k2h þ ðh2 þ k 2 Þ2 : 2 2 Then, (3.1) follows is view of Gronwall’s discrete inequality and (3.4b). h 4. Uniqueness We shall show uniqueness of the approximations U1, U2, . . . , Un satisfying (1.8). Theorem 5. The solution Un of the finite difference scheme (1.8) is unique.
ð3:5Þ
ð3:6Þ
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Proof. Suppose that V0 = u0 and let V n 2 RJ0 þ2 be another solution of (1.8). Then Vn satisfies 1 1 1 ot V n Dh ðot V n Þ þ rh V nþ2 þ u V nþ2 ; V nþ2 ¼ 0:
ð4:1Þ
Let hn = Vn Un with h0 = 0, we obtain from (1.8a) and (4.1) 1 1 1 1 1 ot hn Dh ðot hn Þ þ rh hnþ2 ¼ u U nþ2 ; U nþ2 u V nþ2 ; V nþ2 :
ð4:2Þ
The proof of uniqueness is with the mathematical induction. Now supposing hn = 0 and taking in (4.2) the 1 inner product with hnþ2 , rewriting the terms on the right-hand side according to (3.6), using (1.6) and (1.7) and applying the Schwarz inequality, we obtain h i 2 2 2 2 2 2 ot khn kh þ ot jhn j1;h 6 C jhn j1;h þ jhnþ1 j1;h þ khn kh þ khnþ1 kh : Since h0 = 0, it follows from the discrete Gronwall inequality that for k0 such that 1 Ck > 0 for 0 < k < k0: khnþ1 k1;h ¼ 0: This completes the proof of the uniqueness of solutions for (1.8).
h
5. A linearized difference scheme We construct for (1.8) the following linearized three level difference scheme (see [14–17]): nþ1 ot U nj Dh ot U nj þ rh U j 2 þ u U ^nj ; U ^nj ¼ 0; 1 6 j 6 J ; 1 6 n 6 N 1; U 0j ¼ u0 ðxj Þ; 1 6 j 6 J ; d2 u0 du0 1 1 ðxj Þ þ uðu0 ðxj Þ; u0 ðxj ÞÞ ; U j Dh U j ¼ u0 ðxj Þ 2 ðxj Þ k dx dx
ð5:1aÞ ð5:1bÞ
1 6 j 6 J;
ð5:1cÞ
where U ^n ¼ 32 U n 12 U n1 ; 1 6 n 6 N : For a smooth function u, we have 3 1 1 u^n ¼ un un1 ¼ un2 þ Oðk 2 Þ 2 2
as k ! 0:
ð5:2Þ
Next we need the following Lemma (Sun [20]). Lemma 1. Let b1,b2 and ai, i = 1, 2, 3, . . ., be positive and satisfy aiþ1 6 ð1 þ b1 kÞai þ b2 k; then aiþ1
i ¼ 1; 2; 3; . . . ;
b2 6 expðb1 ikÞ a1 þ ; b1
i ¼ 1; 2; 3; . . .
Theorem 6. Suppose that the solution u(x, t) of (1.1) is sufficiently regular. Then, for k sufficiently small, the solution of linearized modification (5.1) is convergent to the solution of problem (1.1) and the order of convergence rate is O(h2 + k2) in the H1-discrete-norm. Proof. From Taylor expansion, we have nþ1 ot unj Dh ot unj þ rh uj 2 þ u u^nj ; u^nj ¼ P nj ; u0j ¼ u0 ðxj Þ; 2
u1j Dh ðu1j Þ ¼ u0 ðxj Þ
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:3bÞ
d u0 du0 ðxj Þ þ uðu0 ðxj Þ; u0 ðxj ÞÞ þ Qj ; ðxj Þ k 2 dx dx
ð5:3aÞ
1 6 j 6 J;
ð5:3cÞ
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T. Achouri et al. / Applied Mathematics and Computation 182 (2006) 999–1005
where P nj and Qj are the truncation errors of difference scheme (5.1). Using (5.2), there exist a positive constant c0 independent of step sizes such that jP nj j 6 c0 ðh2 þ k 2 Þ;
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:4aÞ
jQj j 6 c0 ðh2 þ k 2 Þ;
1 6 j 6 J:
ð5:4bÞ
Let Enj ¼ unj U nj , we obtain from (5.1) and (5.3) nþ1 ot Enj Dh ot Enj þ rh Ej 2 ¼ u U ^nj ; U ^nj u u^nj ; u^nj þ P nj ; E0j ¼ 0; E1j
1 6 j 6 J ; 1 6 n 6 N 1;
ð5:5bÞ
1 6 j 6 J;
¼ Qj ;
ð5:5aÞ
ð5:5cÞ
1 6 j 6 J:
We will prove by inductive method that kEn k1;h 6 Cðu; T Þðh2 þ k 2 Þ;
ð5:6Þ
0 6 n 6 N:
From (5.5b), we have kE0 k1;h ¼ 0:
ð5:7Þ
Using (5.4b) and (5.5c), we obtain kE1 kh 6 c0 ðh2 þ k 2 Þ:
ð5:8Þ
Now from (5.5c) and Taylor expansion with integration term and by means of (1.1), we get Z 1 sutt ðxj ; k skÞ ds Qj ¼ k 2 0
and rþ h Qj ¼
1 Qjþ1 Qj ¼ k 2 h
Z 0
1
s
utt ðxjþ1 ; k skÞ utt ðxj ; k skÞ ds ¼ k 2 h
Z
1
suttx ðnj ; k skÞ ds;
0
where nj 2 (xj,xj+1). Therefore jE1 j1;h 6 Cðh2 þ k 2 Þ:
ð5:9Þ
Using (5.8) and (5.9) kE1 k1;h 6 Cðh2 þ k 2 Þ:
ð5:10Þ
It follows from (5.7) and (5.10) that (5.6) is valid for n = 0 and n = 1. Now suppose that (5.6) is true for 1 1 6 n 6 l, and taking in (5.5a) the inner product with Enþ2 , we obtain from (1.6) and the boundedness of ux with along line of the proof of Theorem 4 h i 1 2 2 2 2 2 2 ot kEn kh þ ot jEn j1;h 6 C kEn1 k1;h þ kEn k1;h þ kEnþ1 k1;h þ kP n kh : 2 This yields by (5.4a) h i 1 nþ1 2 kE k1;h kEn k21;h 6 C kEn1 k21;h þ kEn k21;h þ kEnþ1 k21;h þ c20 ðh2 þ k 2 Þ2 2k or 2
2
2
2
ð1 2CkÞkEnþ1 k1;h 6 ð1 þ 2CkÞkEn k1;h þ 2CkkEn1 k1;h þ 2c20 kðh2 þ k 2 Þ : 1 Whence, for k 6 6C
kEnþ1 k21;h 6 ð1 þ 6CkÞkEn k21;h þ 3CkkEn1 k21;h þ 3c20 kðh2 þ k 2 Þ2 :
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Here, by above 2 2 2 2 2 max kEnþ1 k1;h ; kEn k1;h 6 ð1 þ 9CkÞ max kEn k1;h ; kEn1 k1;h þ 3c20 kðh2 þ k 2 Þ : By Lemma 1, we have c2 2 2 2 2 2 max kElþ1 k1;h ; kEl k1;h 6 expð9CklÞ maxðkE1 k1;h ; kE0 k1;h Þ þ 0 ðh2 þ k 2 Þ : 3C Using (5.7) and (5.10), we find c2 2 2 2 max kElþ1 k1;h ; kEl k1;h 6 expð9CT Þ C 2 þ 0 ðh2 þ k 2 Þ : 3C This yields kElþ1 k1;h 6 Cðu; T Þðh2 þ k 2 Þ: That means, (5.6) is valid for n = l + 1. This completes the proof.
h
References [1] G.D. Akrivis, Finite difference discretization of the Kuramoto–Sivashinsky equation, Numer. Math. 63 (1992) 1–11. [2] D.N. Arnold, J. Douglas Jr., V. Thome´e, Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable, Math. Comput. 27 (1981) 737–743. [3] T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. Lond. A 272 (1972) 47–78. [4] J.L. Bona, R. Smith, The initial-value problem for the Korteweg–de Vries equation, Philos. Trans. Roy. Soc. Lond. A 278 (1975) 555– 601. [5] F.E. Browder, Existence and uniqueness theorems for solutions of nonlinear boundary value problems, in: Applications of Nonlinear Partial Differential Equations, in: R. Finn (Ed.), Proc. Sympos. Appl. Math., vol. 17, Amer. Math. Soc., Providence, 1965, pp. 24–49. [6] J.C. Eilbeck, G.R. McGuire, Numerical studies of the regularized long-wave equations I: numerical methods, J. Comput. Phys. 19 (1975) 43–57. [7] R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, SIAM J. Numer. Anal. 15 (1978) 1125–1150. [8] R. Kannan, S.K. Chung, Finite difference approximate solutions for the two-dimensional Burger’s system, Comput. Math. Applicat. 44 (2002) 193–200. [9] J.C. Lopez-Marcos, A difference scheme for a nonlinear partial integrodifferential equation, SIAM J. Numer. Anal. 27 (1990) 20–31. [10] L.A. Medeiros, M. Milla Miranda, Weak solutions for a nonlinear dispersive equation, J. Math. Anal. Appl. 59 (1977) 432–441. [11] L.A. Medeiros, G. Perla Menzela, Existence and uniqueness for periodic solutions of the Benjamin–Bona–Mahony equation, SIAM J. Math. Anal 8 (1977) 792–799. [12] M. Milla Miranda, Weak solutions of a modified KdV equation, Bol. Soc. Brasil. Mat. 6 (1975) 57–63. [13] K. Omrani, The convergence of fully discrete Galerkin approximations for the Benjamin–Bona–Mahony (BBM) equation, Appl. Math. Comput., in press, doi:10.1016/j.amc.2005.12.046. [14] K. Omrani, On fully discrete Galerkin approximations for the Cahn–Hilliard equation, Math. Model. Anal. 9 (4) (2004) 313–326. [15] K. Omrani, Convergence of Galerkin Approximations for the Kuramoto–Tsuzuki equation, Numer. Method Partial Differ. Equ. 21 (5) (2005) 961–975. [16] K. Omrani, A second order accurate difference scheme on nonuniform meshes for nonlinear heat-conduction equation, Far East J. Appl. Math. 20 (3) (2005) 355–365. [17] K. Omrani, M.L. Ben Mohamed, A linearized difference scheme for the Sivashinsky equation, Far East J. Appl. Math. 20 (2) (2005) 179–188. [18] D.H. Peregrine, Calculations of the development of an undular bore, J. Fluid Mech. 25 (1966) 321–330. [19] M.A. Raupp, Galerkin methods applied to the Benjamin–Bona–Mahony equation, Bol. Soc. Brasil. Mat. 6 (1975) 65–77. [20] Z.Z. Sun, A second-order accurate linearized difference scheme for the two-dimensional Cahn–Hilliard equation, Math. Comput. 64 (212) (1995) 1463–1471. [21] L. Wahlbin, Error estimates for a Galerkin method for a class of model equations for long waves, Numer. Math. 23 (1975) 289–303.