ERROR ESTIMATES OF NONSTANDARD FINITE DIFFERENCE SCHEMES FOR GENERALIZED CAHN-HILLIARD AND KURAMOTO-SIVASHINSKY EQUATIONS
Sang Mok Choo ∗ , Sang Kwon Chung, and Yoon Ju Lee
Abstract. Nonstandard finite difference schemes are considered for a generalization of the Cahn-Hilliard equation with Neumann boundary conditions and the Kuramoto-Sivashinsky equation with periodic boundary conditions, which are of the type
u
t+ ∂
2∂x
2g(u, u
x, u
xx) = ∂
α∂x
αf (u, u
x), α = 0, 1, 2.
Stability and error estimate of approximate solutions for the corre- sponding schemes are obtained using the extended Lax-Richtmyer equivalence theorem. Three examples are provided to apply the nonstandard finite difference schemes.
1. Introduction
Consider the partial differential equation (1.1) u t + ∂ 2
∂x 2 g(u, u x , u xx ) = ∂ α
∂x α f (u, u x ), x ∈ Ω, 0 < t ≤ T, with an initial condition
(1.2) u(x, 0) = u 0 (x), x ∈ Ω, and either Neumann boundary conditions
(1.3a) ∂u
∂x = 0, ∂ 3 u
∂x 3 = 0, (x, t) ∈ ∂Ω × (0, T ],
Received May 3, 2004.
2000 Mathematics Subject Classification: 35G25, 65M06, 65M12, 65M15.
Key words and phrases: nonstandard finite difference scheme, Cahn-Hilliard equa- tion, Kuramoto-Sivashinsky equation, Neumann boundary condition, periodic bound- ary condition, Lax-Richtmyer equivalence theorem.
∗
This work was supported by the Korea Research Foundation Grant (KRF-2002-
041-C00034).
or a periodic boundary condition
(1.3b) u(x, t) = u(x + 1, t), (x, t) ∈ Ω × (0, T ].
Here, α = 0, 1, 2, Ω = (0, 1) and f and g are given functions. We assume that g(u, u x , u xx ) = g 1 (u, u x )u xx + g 2 (u, u x ) and there exist constants G 0 and G 1 such that 0 < G 0 ≤ g 1 (x, y) ≤ G 1 for all real x and y. The regularity of g and f is given in Theorems 3.3–3.4. In the case of using the Neumann boundary conditions (1.3a), we also assume g 1 (x, −y) = g 1 (x, y) and g 2 (x, −y) = g 2 (x, y) for all x, y, in order to use an integration by parts in the proof of Theorem 3.3.
In case of α = 2, the partial differential equation (1.1)–(1.2) can be found as a generalization of the Cahn-Hilliard equation with g 1 (u, u x ) = q, g 2 (u, u x ) = 0, f (u, u x ) = −pu + su 3 and the Neumann boundary con- ditions (1.3a). Here, p, q and s are positive constants. Global existence and uniqueness of the solution for the Cahn-Hilliard equation have been shown by Elliott and Zheng[9]. Finite element Galerkin approximate solutions have been obtained by Elliott and French[7]–[8]. Furihata[10]–
[11] has proposed a finite difference scheme which has inherited the de- crease of the total energy. Choo and Chung[4] and Choo, Chung and Kim[5] have considered a conservative nonlinear difference scheme and obtained the corresponding error estimates.
In case of α = 0, the equations (1.1)–(1.2) with the periodic bound- ary condition (1.3b) can be considered as the Kuramoto-Sivashinsky equation with g 1 (u, u x ) = 1, g 2 (u, u x ) = −u, f (u, u x ) = 1 2 u 2 x . Existence and uniqueness of the solution for the Kuramoto-Sivashinsky equation have been shown by Tadmor[16]. Finite difference schemes and finite element Galerkin methods have been applied by Akrivis([2],[3]). For the Kuramoto-Sivashinsky equation with Dirichlet boundary conditions, Manickam, Moudgalya and Pani[13] have applied a second order split- ting method with orthogonal cubic spline collocation methods.
In this paper, we consider error estimates of approximate solutions
for nonstandard finite difference methods. It is necessary to introduce
nonstandard finite difference schemes in order to preserve positivity and
energy decay of analytical solutions(see Mickens[14]). In Section 2, we
introduce the general nonstandard finite difference scheme for (1.1)–(1.2)
with either the boundary conditions (1.3a) or (1.3b). Some preliminary
lemmas and discrete norms are given. In Sections 3, we briefly recall
the Lax-Richtmyer equivalence theorem and obtain stability and error
estimates for the difference equation by following the idea in Lopez-
Marcos and Sanz-Serna[12]. In Section 4, we give specific examples.
2. Nonstandard finite difference scheme
Let h = M 1 be the uniform step size in the spatial direction for a positive integer M and Ω h = {x i = ih|i = −2, −1, 0, · · · , M, M + 1, M + 2}. Let k = N T denote the uniform step size in the temporal direction for a positive integer N . Denote V i n = V (x i , t n ) for t n = nk, n = 0, 1, · · · , N. For a function V n = (V −2 n , V −1 n , V 0 n , · · · , V M n , V M +1 n , V M +2 n ) defined on Ω h , define the difference operators as for 0 ≤ i ≤ M ,
∇ + V i n = V i+1 n − V i n
h , ∇ − V i n = V i n − V i−1 n
h , ¯ ∇V i n = 1
2 (∇ + + ∇ − )V i n ,
∇ 2 V i n = ∇ + (∇ − V i n ) and ∇ 4 V i n = ∇ 2 (∇ 2 V i n ).
Further, define operators V n+1/2 and ∂ t V n , respectively, as V i n+1/2 = V i n+1 + V i n
2 and ∂ t V i n = V i n+1 − V i n
k .
Then the approximate solution U n for (1.1)–(1.2) is defined as a solution of
∂ t U i n +∇ 2 n
g 1
³
U i n+1/2 , ¯ ∇U i n+1/2
´
∇ 2 U i n+1/2 o
= − ∇ 2 g 2
³
U i n+1/2 , ¯ ∇U i n+1/2
´
+ ∇ α P
³
U i n , U i n+1 , ∇ − U i n+1/2 , ∇ + U i n+1/2
´ (2.1)
with the initial condition
(2.2) U i 0 = u 0 (x i ), 0 ≤ i ≤ M and the boundary conditions either
(2.3a) ∇U ¯ i n = 0, ¯ ∇∇ 2 U i n = 0, i = 0, M, 1 ≤ n ≤ N or
(2.3b) U i n = U i+M n , −∞ < i < ∞, 0 ≤ n ≤ N.
Here, ∇ 1 = ¯ ∇ and P may be any C α+2 (R 4 )-function satisfying P (u n i , u n+1 i , ∇ − u n+1/2 i , ∇ + u n+1/2 i ) = f (u n+1/2 i , u n+1/2 i,x ) + O(k γ + h ζ ) with 1 < γ and 1 ≤ ζ. In the case of using the Neumann boundary conditions (1.3a) and α = 2, P also must satisfy
(2.4) P (x, y, −z, −w) = P (x, y, w, z), ∀x, y, z, w
in order to use an integration by parts in the proof of Theorem 3.3.
For simplicity, we denote P
³
U i n , U i n+1 , ∇ − U i n+1/2 , ∇ + U i n+1/2
´ by [P U ] n+1/2 i . Note that the discretized Neumann boundary conditions (2.3a) is equal to U −1 n = U 1 n , U M +1 n = U M −1 n , U −2 n = U 2 n and U M +2 n = U M −2 n .
In order to consider the error estimates corresponding to the finite difference equation (2.1) with the initial condition (2.2) and the bound- ary condition (2.3a), we now introduce the discrete L 2 -inner product
(V, W ) h = h X M 00
i=0
V i W i = h n 1
2 (V 0 W 0 + V M W M ) +
M −1 X
i=1
V i W i o
and the corresponding discrete L 2 -norm kV k h = (V, V ) h12
for functions V = (V −2 , · · · , V M +2 ) and W = (W −2 , · · · , W M +2 ) satisfy- ing (2.3a). For the maximum norm, we define
kV k ∞ = max
0≤i≤M |V i |.
Hereafter, whenever there is no confusion, (·, ·) and k·k will denote (·, ·) h and k · k h , respectively.
It follows from summation by parts that the following Lemma 2.1 holds.
Lemma 2.1. For functions V and W defined on Ω h and satisfying the first equality in (2.3a), the following identity and inequalities hold.
(1) (∇ 2 V, W ) = −h P M
i=1 (∇ − V i )(∇ − W i ) = (V, ∇ 2 W ).
(2) k∇ + V k 2 ≤ 2kV kk∇ 2 V k.
(3) k∇ − V k 2 ≤ 2kV kk∇ 2 V k.
The following lemma can be verified by summation by parts and using minimum eigenvalue of a symmetric matrix. For a proof, see Agarwal[1].
Lemma 2.2. Let M be any positive integer. If V 0 = 0, then
½
2 sin π 2(2M + 1)
¾ 2 M X
i=1
V i 2 ≤ X M
i=1
(V i − V i−1 ) 2 .
Lemma 2.3. For a function V defined on Ω h , the inverse inequality k ¯ ∇V k ≤ 1
h kV k
holds.
Proof. From the definition of the discrete norm, it holds that k ¯ ∇V k 2 = h
X M 00 i=0
³ V i+1 − V i−1 2h
´ 2
≤ 2 4h
X M 00 i=0
(V i+1 2 + V i−1 2 )
= 1 2h
·
V 0 2 + V M 2 + 2
M −1 X
i=1
V i 2
¸
= 1 h 2 kV k 2 . Thus, we obtain the inverse inequality.
Using Lemma 2.2 and the definition of ¯ ∇( ¯ ∇), we obtain the following inequalities.
Lemma 2.4. Let V be a function defined on Ω h and satisfying (2.3a).
Then
(1) k ¯ ∇V k ≤ 2 √
2k∇ 2 V k.
(2) k ¯ ∇( ¯ ∇V )k ≤ 3 √ 4 2 k∇ 2 V k.
It follows from Lemma 2.3 and Lemma 2.4 that the following lemma holds.
Lemma 2.5. Let V be a function defined on Ω h and satisfying (2.3a).
Then
(1) kV k 2 ∞ ≤ 3kV k 2 + 8kV kk ¯ ∇V k.
(2) k ¯ ∇V k 2 ∞ ≤ 5k ¯ ∇V k 2 + 9 4 k∇ 2 V k 2 . (3) k∇ 2 V k 2 ∞ ≤ (3 + h 8 )k∇ 2 V k 2 .
Proof. Let µ be an odd index such that
V µ 2 = min{V 2`+1 2 | 0 ≤ 2` + 1 ≤ M, ` is an integer}.
Then
V µ 2 = h X M 00
i=0
V µ 2 ≤ 3h X M 00
i=0
V i 2 = 3kV k 2 . For all positive integers j such that µ + 2j ≤ M , we obtain
V µ+2j 2 − V µ 2 = X j−1
i=0
{V 2(i+1)+µ + V 2i+µ }{V 2(i+1)+µ − V 2i+µ }
= 2h
j−1 X
i=0
{V 2(i+1)+µ + V 2i+µ } ¯ ∇V 2i+1+µ
≤ 8kV kk ¯ ∇V k.
This implies
V µ+2j 2 ≤ V µ 2 + 8kV kk ¯ ∇V k ≤ 3kV k 2 + 8kV kk ¯ ∇V k.
Similarly, we obtain for all positive integers j such that µ − 2j ≥ 0.
V µ−2j 2 ≤ V µ 2 + 8kV kk ¯ ∇V k ≤ 3kV k 2 + 8kV kk ¯ ∇V k.
Letting ν be an even index such that
V ν 2 = min{V 2` 2 | 0 ≤ 2` ≤ M, ` is an integer}
and following the above idea, we obtain
V ν+2j 2 ≤ 3kV k 2 + 8kV kk ¯ ∇V k and V ν−2j 2 ≤ 3kV k 2 + 8kV kk ¯ ∇V k.
Thus, the first inequality (1) is proved.
The second inequality (2) follows from the inequality (1) and Lemma 2.4.
Since we obtain
kV k 2 ∞ ≤ (3 + 8 h )kV k 2
from the inequality (1) and Lemma 2.3, the inequality (3) is proved.
Remark 2.1. For V and W defined on Ω h and satisfying V i = V M +i and W i = W M +i with i = 0, 1, we obtain
(∇ 2 V, W ) = (V, ∇ 2 W ).
And for the periodic boundary condition (2.3b), we also obtain similar results in Lemmas 2.1–2.5 which are sufficient for the proof of Theorem 3.3. For proofs, we refer to Ortega and Sanz-Serna[15].
3. Convergence of approximate solution
We recall the extension of Lax-Richtmyer equivalence theorem in Lopez-Marcos and Sanz-Serna[12] which makes us avoid the difficulty of direct proof for convergence arising specially in nonlinear problems.
Let u be a solution of a problem Φ(u) = 0 and u h be a discrete eval- uation of u on Ω h . Let U h be an approximate solution of u, which is obtained by solving the discrete equation
(3.1) Φ h (U h ) = 0,
where Φ h : X h → Y h is a continuous mapping and X h , Y h are normed
spaces having the same dimension. The scheme (3.1) is said to be con-
vergent if (3.1) has a solution U h such that lim h→0 kU h −u h k Xh= 0. The
discretization (3.1) is said to be consistent if lim h→0 kΦ h (u h )k Yh = 0.
= 0.
The scheme (3.1) is said to be stable in the threshold R h if there exists a positive constant C such that for an open ball B(u h , R h ) ⊂ X h ,
kV h − W h k Xh≤ CkΦ h (V h ) − Φ h (W h )k Yh, ∀ V h , W h ∈ B(u h , R h ).
, ∀ V h , W h ∈ B(u h , R h ).
The following theorem is the extended Lax-Richtmyer equivalence theorem which gives existence and convergence of approximate solutions.
For the proof, see [12].
Theorem 3.1. Assume that the discrete equation (3.1) is consistent and stable in the threshold R h . If Φ h is continuous in B(u h , R h ) and kΦ h (u h )k Yh = o(R h ) as h → 0, then (3.1) has a unique solution U h in B(u h , R h ) and there exists a constant C such that
kU h − u h k Xh ≤ CkΦ h (u h )k Yh.
.
According to Theorem 3.1, we have only to show that (2.1) is consis- tent and stable in the threshold in order to show the unique existence and convergence of approximate solutions.
Let Z h n be the set of all functions defined on Ω h satisfying the dis- cretized Neumann boundary condition (2.3a) at time level n (0 ≤ n ≤ N ). We take X h = Y h = Q N
n=0 Z h n and define a mapping Φ h : X h → Y h by Φ h (U) = ˜ U, where for n = 0, · · · , N − 1,
U ˜ i n+1 = ∂ t U i n + ∇ 2 n
g 1
³
U i n+1/2 , ¯ ∇U i n+1/2
´
∇ 2 U i n+1/2 o
+ ∇ 2 g 2
³
U i n+1/2 , ¯ ∇U i n+1/2
´
− ∇ α [P U ] n+1/2 i (3.2)
and
(3.3) U ˜ 0 = U 0 − u 0 .
We take norms k · k Xh and k · k Yh on X h and Y h , respectively, such that
on X h and Y h , respectively, such that
kUk 2 Xh= max
0≤n≤N kU n k 2 + k
N −1 X
n=0
k∇ 2 U n+1/2 k 2 and
k ˜ Uk 2 Yh= k ˜ U 0 k 2 + k X N n=1
k ˜ U n k 2 .
The consistency of the scheme (2.1)–(2.2) is obtained using Taylor’s
Theorem and the Mean Value Theorem.
Theorem 3.2. Let u be the solution of (1.1)–(1.2) with bounded derivatives ∂ ∂t3u
3, ∂ ∂x6u
6 and u h be the discretized evaluation of u. Assume g 1 , g 2 ∈ C 4 (R 2 ), f ∈ C α+2 (R 2 ) and P ∈ C α+2 (R 4 ). Then there exists a constant C such that for 1 < γ and 1 ≤ ζ,
u
6and u h be the discretized evaluation of u. Assume g 1 , g 2 ∈ C 4 (R 2 ), f ∈ C α+2 (R 2 ) and P ∈ C α+2 (R 4 ). Then there exists a constant C such that for 1 < γ and 1 ≤ ζ,
kΦ h (u h )k Yh ≤ C(k γ + h ζ ).
We now consider the stability of the approximate solution in the threshold R h .
Theorem 3.3. Let Φ h (U) = ˜ U and Φ h (V) = ˜ V. Assume that u xx ∈ L ∞ (Ω×(0, T )), g 1 , g 2 , f ∈ C 1 (R 2 ), P ∈ C 1 (R 4 ) and R h = O(h r k s ) with r ≥ 1 2 and s ≥ 1 2 . Then there exists a constant C such that for any U and V in the ball B(u h , R h ),
kU − Vk Xh ≤ CkΦ h (U) − Φ h (V)k Yh.
.
Proof. Let e n = U n − V n and ˜ K n = ˜ U n − ˜ V n . Replacing U n and U ˜ n in (3.2) by V n and ˜ V n , respectively, and subtracting this result from (3.2), we obtain
∂ t e n i + ∇ 2 n
g 1
³
U i n+1/2 , ¯ ∇U i n+1/2
´
∇ 2 e n+1/2 i o
= ˜ K i n+1 + ∇ 2
hn g 1
³
V i n+1/2 , ¯ ∇V i n+1/2
´
− g 1
³
U i n+1/2 , ¯ ∇U i n+1/2
´o
∇ 2 V i n+1/2 i
− ∇ 2 n
g 2
³
U i n+1/2 , ¯ ∇U i n+1/2
´
− g 2
³
V i n+1/2 , ¯ ∇V i n+1/2
´o
+ ∇ α
³
[P U ] n+1/2 i − [P V ] n+1/2 i
´ . (3.4)
It follows from Lemma 2.5, the definition of k · k Xh and R h = O(h r k s ) with r ≥ 1 2 and s ≥ 1 2 that for V in the ball B(u h , R h ),
° °
°∇ 2 V n+1/2
° °
° ∞ ≤
° °
°∇ 2
³
V n+1/2 − u n+1/2 h
´° °
° ∞ +
° °
°∇ 2 u n+1/2 h
° °
° ∞
≤ C µ
1 + 1 h 1/2
¶ h r k s k 1/2 + C
≤ C (
1 + k s−1/2 Ã 1
X
`=0
h r−`/2
!)
.
(3.5)
Using the Mean Value Theorem, Lemma 2.1 and (3.5), we obtain for some 0 < ξ 1 < 1 and 0 < ξ 2 < 1,
° °
° n
g 1
³
U n+1/2 , ¯ ∇U n+1/2
´
− g 1
³
V n+1/2 , ¯ ∇V n+1/2
´o
× ∇ 2 V n+1/2
° °
°
° °
°∇ 2 e n+1/2
° °
°
≤
° °
°∇ 2 V n+1/2
° °
° ∞
n° °
°g 1,x
³
ξ 1 U n+1/2 + (1 − ξ 1 )V n+1/2 , ¯ ∇U n+1/2
´° °
° ∞
×
° °
°e n+1/2
° °
° +
° °
°g 1,y
³
V n+1/2 , ¯ ∇ξ 2 U n+1/2 + ¯ ∇(1 − ξ 2 )V n+1/2
´° ° °
∞
×
° °
° ¯ ∇e n+1/2
° °
° o° °
°∇ 2 e n+1/2
° °
°
≤ C
° °
°∇ 2 V n+1/2
° °
° ∞
³° °
°e n+1/2
° °
° +
° °
° ¯ ∇e n+1/2
° °
°
´° °
°∇ 2 e n+1/2
° °
°
≤ C
° °
°∇ 2 V n+1/2
° °
° ∞
³° ° °e n+1/2
° °
° +
° °
°∇ − e n+1/2
° °
° +
° °
°∇ + e n+1/2
° °
°
´° ° °∇ 2 e n+1/2
° °
°
≤ C
° °
°∇ 2 V n+1/2
° °
° ∞
³° °
°e n+1/2
° °
° +
° °
°e n+1/2
° °
°
1 2
°
° °∇ 2 e n+1/2
° °
°
1 2
´° °
°∇ 2 e n+1/2
° °
°
≤ C
³° °
°∇ 2 V n+1/2
° °
° 2
∞ +
° °
°∇ 2 V n+1/2
° °
° 4
∞
´° °
°e n+1/2
° °
° 2 + G 0 4
° °
°∇ 2 e n+1/2
° °
° 2
≤ C n
1 + X 2
i=0
k (s−1/2)2i
³ X 1
`=0
h r−`/2
´ 2io°
° °e n+1/2
° °
° 2 + G 0 4
° °
°∇ 2 e n+1/2
° °
° 2 . (3.6)
Further, since [P V ] n+1/2 = P (V n , V n+1 , ∇ − V n+1/2 , ∇ + V n+1/2 ), it follows from the Mean Value Theorem and Lemma 2.1 that for α = 0, 1,
³
∇ α n
[P U ] n+1/2 − [P V ] n+1/2 o
, e n+1/2
´
≤ C
½° °
°e n
° °
° +
° °
°e n+1
° °
° +
° °
°e n+1/2
° °
° 1/2
° °
°∇ 2 e n+1/2
° °
° 1/2
¾ ° °
°e n+1/2
° °
° (3.7)
and for α = 2,
³
∇ α n
[P U ] n+1/2 − [P V ] n+1/2 o
, e n+1/2
´
≤ C
½° ° °e n
° °
° +
° °
°e n+1
° °
° +
° °
°e n+1/2
° °
° 1/2
° °
°∇ 2 e n+1/2
° °
° 1/2
¾ ° ° °∇ 2 e n+1/2
° °
°.
(3.8)
Taking an inner product between (3.4) and 2e n+1/2 , and applying Lemma 2.1, (3.6) and (3.7)–(3.8), we obtain
∂ t ke n k 2 + 2
° °
° q
g 1 (U n+1/2 , ¯ ∇U n+1/2 )∇ 2 e n+1/2
° °
° 2
≤ k ˜ K n+1 k 2 + ke n+1/2 k 2 + C
1 + X 2
i=0
k (s−1/2)2i
à 1
X
`=0
h r−`/2
! 2i
(ke n k 2 + ke n+1 k 2 ) + G 0
2 k∇ 2 e n+1/2 k 2 + C(ke n k 2 + ke n+1 k 2 ) + G 0
2 k∇ 2 e n+1/2 k 2 . Thus, we have
∂ t ke n k 2 + G 0 k∇ 2 e n+1/2 k 2
≤ k ˜ K n+1 k 2 + C n
1 + X 2 i=0
k (s−1/2)2i( X 1
`=0
h r−`/2 ) 2i) o¡
ke n k 2 + ke n+1 k 2 ¢ . (3.9)
Applying the discrete Gronwall’s inequality to (3.9), we obtain for m ≥ 0,
ke m+1 k 2 + k X m n=0
k∇ 2 e n+1/2 k 2 ≤ S n
ke 0 k 2 + k
m+1 X
n=1
k ˜ K n k 2 o
,
where S = exp µ C ©
1+ P
2i=0
k
(s−1/2)2i( P
1`=0
h
r−`/2)
2iª
1−kC ©
1+ P
2i=0
k
(s−1/2)2i( P
1`=0
