# A Fourth-Order Conservative Compact Finite Difference Scheme for the Generalized RLW Equation

Authors: Shuguang Li; Jue Wang; Yuesheng Luo
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

The generalized regularized long-wave (GRLW) equation is studied by finite difference method. A new fourth-order energy conservative compact finite difference scheme was proposed. It is proved by the discrete energy method that the compact scheme is solvable, the convergence and stability of the difference schemes are obtained, and its numerical convergence order isO
(
τ
2
+
h
4
) in the L
∞-norm. Further, the compact schemes are conservative. Numerical experiment results show that the theory is accurate and the method is efficient and reliable.


---

## Body

## 1. Introduction

In this paper, we consider the following generalized regularized long-wave equation:(1)
u
t
+
u
x
+
α
u
p
x
-
β
u
x
x
t
=
0
,
(
x
,
t
)
∈
(
x
l
,
x
r
)
×
(
0
,
T
]
,with the boundary conditions (2)
u
(
x
l
,
t
)
=
u
(
x
r
,
t
)
=
0
,
t
∈
(
0
,
T
]
,and the initial condition (3)
u
(
x
,
0
)
=
u
0
(
x
)
,
x
∈
(
x
l
,
x
r
)
,where u
=
u
(
x
,
t
) is a real-valued function defined on (
x
l
,
x
r
)
×
(
0
,
T
], α
,
β
>
0, p
≥
2 is a positive integer, and u
0 is a given function with Dirichlet boundary condition.The GRLW equation was first put forward as a model for small-amplitude long waves on the surface of water in a channel by Peregrine [1, 2]. A special case of (1), that is, (4)
u
t
+
u
x
+
α
u
u
x
-
β
u
x
x
t
=
0
,is usually called the regularized long-wave (RLW) equation. The RLW equation is a representation form of nonlinear long wave and can describe a lot of important physical phenomena, such as shallow waves and ionic waves. The GRLW equation can also describe that wave motion to the same order of approximation as the KDV equation, so it plays a major role in the study of nonlinear dispersive waves [3]. It is difficult to find the analytical solution for (1), which has been studied by many researchers. The finite difference method for the initial-boundary value problem of the GRLW equation had been studied in [4–8]. Other mathematical theory and numerical methods for GRLW equation were considered in [9–11]. Reference [12] solved the GRLW equation by the Petrov-Galerkin method. Numerical solution of GRLW equation used Sinc-collocation method in [13]. In [14], a time-linearization method that uses a Crank-Nicolson procedure in time and three-point, fourth-order accurate in space, compact difference equations, is presented and used to determine the solutions of the generalized regularized-long wave (GRLW) equation. Recently, there has been growing interest in high-order compact methods for solving partial differential equations [15–17].In this paper, we consider problem (1)–(3); it has the following conservation law: (5)
E
(
t
)
=
u
L
2
2
+
β
u
x
L
2
2
=
const
.Using a customary designation, we will refer to the functional E
(
t
) as the energy integral, although it is not necessarily identifiable with energy in the original physical problem.We aim to present a conservative finite difference scheme for problem (1)–(3), which simulates conservation law (5) that the differential equation (1) possesses, and prove convergence and stability of the scheme. This paper is organized as follows. In Section 2, some notations are given and some useful lemmas are proposed. In Section 3, we present a nonlinear compact conservative difference scheme, discuss its discrete conservative law, prove the existence of difference solution by Brouwer fixed point theorem, give some a priori estimates, and then prove by discrete energy method that the difference scheme is uniquely solvable, unconditionally stable and that convergence of the difference solutions with O
(
τ
2
+
h
4
) order is based on some a priori estimates. In Section 4, numerical results are provided to test the theoretical results.


## 2. Notations and Lemmas

Leth
=
(
x
r
-
x
l
)
/
J and τ
=
T
/
N be the spatial and temporal step sizes, respectively. Denote x
j
=
x
l
+
j
h
,
(
0
≤
j
≤
J
)
,
t
n
=
n
τ
,
(
0
≤
n
≤
N
). Let u
j
n denote the approximation of u
(
x
j
,
t
n
), and let (6)
R
0
J
=
{
u
j
=
u
j
j
∈
Z
∣
u
0
=
u
J
=
0
}
.As usual, the following notations will be used:(7)
δ
x
+
u
j
n
=
u
j
+
1
n
-
u
j
n
h
,
δ
x
-
u
j
n
=
u
j
n
-
u
j
-
1
n
h
,
δ
x
u
j
n
=
u
j
+
1
n
-
u
j
-
1
n
2
h
,
u
-
j
n
=
u
j
n
+
1
+
u
j
n
-
1
2
,
δ
t
u
j
n
=
u
j
n
+
1
-
u
j
n
-
1
2
τ
,
A
1
u
j
n
=
(
1
+
h
2
12
δ
x
+
δ
x
-
)
u
j
n
,
A
2
u
j
n
=
(
1
+
h
2
6
δ
x
+
δ
x
-
)
u
j
n
.We now introduce the discreteL
2-inner product and the associated norm (8)
u
n
,
v
n
h
=
h
∑
j
=
1
J
-
1
‍
u
j
n
v
j
n
,
u
n
h
=
u
n
,
u
n
h
1
/
2
.The discreteL
∞-norm ·
∞
,
h is defined as (9)
u
∞
,
h
=
max
⁡
j
u
j
.LetΩ
h
=
{
x
j
=
x
l
+
j
h
∣
0
≤
j
≤
J
}. It is convenient to let L
h
2
(
Ω
h
) denote the normed vector space as {
R
0
J
,
·
h
}. The corresponding matrices are defined, respectively, as(10)
u
n
=
u
1
n
,
u
2
n
,
…
,
u
J
-
1
n
T
,
u
n
p
=
diag
(
u
1
n
p
,
u
2
n
p
,
…
,
u
J
-
1
n
p
)
,
A
1
=
1
12
10
1
0
⋯
0
1
10
1
⋯
0
⋱
⋱
⋱
0
⋯
1
10
1
0
⋯
0
1
10
J
-
1
,
A
2
=
1
6
4
1
0
⋯
0
1
4
1
⋯
0
⋱
⋱
⋱
0
⋯
1
4
1
0
⋯
0
1
4
J
-
1
.For a simple notation, the discrete functionφ is defined by (11)
φ
(
u
,
v
)
=
p
α
p
+
1
[
u
p
-
1
H
2
δ
x
v
+
H
2
δ
x
(
u
p
-
1
v
)
]
,where H
1
=
A
1
-
1
,
H
2
=
A
2
-
1. Obviously, A
1, A
2, H
1, H
2 are symmetric positive definite matrices. To obtain some important results, we introduce the following lemmas.Lemma 1 (see [18]).
Foru
,
v
∈
R
0
J, one has (12)
δ
x
+
u
,
v
h
=
-
u
,
δ
x
-
v
h
,
δ
x
u
,
v
h
=
-
u
,
δ
x
v
h
.Lemma 2.
For any real symmetric positive definite matricesH and for u
,
v
∈
R
0
J, one can get (13)
H
δ
x
+
δ
x
-
u
,
v
h
=
-
H
δ
x
+
u
,
δ
x
+
v
h
=
-
R
δ
x
+
u
,
R
δ
x
+
v
h
,
H
δ
x
u
,
v
h
=
-
H
u
,
δ
x
v
h
=
-
u
,
H
δ
x
v
h
,where R is obtained by Cholesky decomposition of H, denoted as H
=
R
T
R.Proof.
Foru
,
v
∈
R
0
J, we have (14)
H
δ
x
+
δ
x
-
u
,
v
h
=
δ
x
+
δ
x
-
H
u
,
v
h
=
-
δ
x
+
H
u
,
δ
x
+
v
h
=
-
(
H
δ
x
+
u
,
δ
x
+
v
)
h
=
-
R
T
R
δ
x
+
u
,
δ
x
+
v
h
=
-
R
δ
x
+
u
,
R
δ
x
+
v
h
,
H
δ
x
u
,
v
h
=
δ
x
H
u
,
v
h
=
-
(
H
u
,
δ
x
v
)
h
=
-
u
,
H
δ
x
v
h
.Lemma 3 (see [16]).
On the matricesA
1, A
2. The eigenvalues of the matrices A
1 and A
2 are, respectively, as follows: (15)
λ
A
1
,
i
=
1
6
5
+
cos
⁡
2
i
π
J
,
λ
A
2
,
i
=
1
3
2
+
cos
⁡
2
i
π
J
,
i
=
1,2
,
…
,
J
-
1
.Lemma 4.
Foru
∈
R
0
J, we can get (16)
u
h
2
≤
H
1
u
,
u
h
=
R
1
u
h
2
≤
3
2
u
h
2
,
u
h
2
≤
H
2
u
,
u
h
=
R
2
u
h
2
≤
3
u
h
2
,where R
l are obtained by Cholesky decomposition of H
l, denoted as H
l
=
R
l
T
R
l
,
(
l
=
1,2
).Proof.
It follows from Lemma3 that the eigenvalues of H
1 and H
2 satisfy (17)
1
≤
λ
H
1
,
k
≤
3
2
,
k
=
1,2
,
…
,
J
-
1
,
1
≤
λ
H
2
,
k
≤
3
,
k
=
1,2
,
…
,
J
-
1
.This gives the spectral radius ρ
(
H
1
)
≤
3
/
2,  ρ
(
H
2
)
≤
3, and consequently (18)
1
≤
H
1
=
ρ
(
H
1
)
≤
3
2
,
1
≤
H
2
=
ρ
(
H
2
)
≤
3
.Thus (19)
u
h
2
≤
H
1
u
,
u
h
=
R
1
u
,
R
1
u
h
≤
H
1
u
,
u
h
≤
3
2
u
h
2
,
u
h
2
≤
(
H
2
u
,
u
)
h
=
(
R
2
u
,
R
2
u
)
h
≤
H
2
u
,
u
h
≤
3
u
h
2
.Lemma 5.
Foru
,
v
∈
R
0
J, one has (20)
φ
u
,
v
,
v
h
=
0
.Proof.
Foru
∈
R
0
J, we have (21)
φ
u
,
v
,
v
h
=
p
α
p
+
1
u
p
-
1
H
2
δ
x
v
+
H
2
δ
x
u
p
-
1
v
,
v
h
=
p
α
p
+
1
[
u
p
-
1
H
2
δ
x
v
,
v
h
+
H
2
δ
x
u
p
-
1
v
,
v
h
]
=
p
α
p
+
1
[
H
2
δ
x
v
,
u
p
-
1
v
h
-
u
p
-
1
v
,
H
2
δ
x
v
h
]
=
0
.Lemma 6 (see [18]).
For any discrete functionu
∈
R
0
J and for any given ε
>
0, there exists a constant K
(
ε
,
n
), depending only on ε and n, such that (22)
u
n
∞
,
h
≤
ε
δ
x
+
u
n
h
+
K
(
ε
,
n
)
u
n
h
.Lemma 7 (see [19]).
Let(
H
,
(
·
,
·
)
) be a finite-dimensional inner product space, let · be the associated norm, and let g
:
H
→
H be continuous. Assume, moreover, that ∃
α
>
0, ∀
z
∈
H, z
=
α, (
g
(
z
)
,
z
)
>
0. Then, there exists a z
*
∈
H such that g
(
z
*
)
=
0 and z
*
≤
α.Lemma 8 (see [18]).
Suppose that the discrete functionw
h satisfies recurrence formula (23)
w
n
-
w
n
-
1
≤
A
τ
w
n
+
B
τ
w
n
-
1
+
C
n
τ
,where A, B, and C
n
(
n
=
1
,
…
,
N
) are nonnegative constants. Then (24)
w
h
≤
w
0
+
τ
∑
k
=
1
N
‍
C
k
e
2
(
A
+
B
)
τ
,where τ is sufficiently small, such that (
A
+
B
)
τ
≤
N
-
1
/
2
N
(
N
>
1
).


## 3. A Nonlinear-Implicit Conservative Scheme

In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1)–(3) and give its numerical analysis.


### 3.1. The Nonlinear-Implicit Scheme and Its Conservative Law

Next we consider the compact finite difference scheme for problem (1)–(3) as follows: (25)
∂
t
u
j
n
+
θ
A
2
-
1
δ
x
u
j
n
+
(
1
-
θ
)
A
2
-
1
δ
x
u
-
j
n
+
p
α
p
+
1
[
u
-
j
n
p
-
1
A
2
-
1
δ
x
u
-
j
n
+
A
2
-
1
δ
x
u
-
j
n
p
]
-
β
A
1
-
1
δ
x
+
δ
x
-
(
∂
t
u
j
n
)
=
0
,
1
≤
j
≤
J
-
1
,
1
≤
n
≤
N
-
1
,
u
0
n
=
u
J
n
=
0
,
1
≤
n
≤
N
,
u
j
0
=
u
0
(
x
j
)
,
0
≤
j
≤
J
,where weight coefficient θ
∈
[
0,1
]. Note that we need another finite difference scheme to calculate u
1, so the following scheme will be used:(26)
u
j
1
-
β
A
1
-
1
δ
x
+
δ
x
-
u
j
1
=
u
0
(
x
j
)
-
β
∂
2
u
0
∂
x
2
(
x
j
)
-
τ
∂
u
0
∂
x
(
x
j
)
-
τ
u
0
(
x
j
)
∂
u
0
∂
x
(
x
j
)
.The matrix form of the difference scheme (25) can be written as(27)
∂
t
u
n
+
θ
H
2
δ
x
u
n
+
(
1
-
θ
)
H
2
δ
x
u
-
n
+
p
α
p
+
1
[
u
-
n
p
-
1
H
2
δ
x
u
-
n
+
H
2
δ
x
u
-
n
p
]
-
β
H
1
δ
x
+
δ
x
-
(
∂
t
u
n
)
=
0
,
1
≤
n
≤
N
-
1
,
(28)
u
0
n
=
u
J
n
=
0
,
1
≤
n
≤
N
,
(29)
u
j
0
=
u
0
(
x
j
)
,
0
≤
j
≤
J
.Theorem 9.
Suppose thatu
0
∈
H
0
1
(
Ω
); then the finite difference scheme (25) is conservative for discrete energy; that is, (30)
E
n
=
1
2
u
n
+
1
h
2
+
u
n
h
2
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
+
τ
θ
R
2
δ
x
u
n
,
R
2
u
n
+
1
=
E
n
-
1
=
⋯
=
E
0
,where R
l are obtained by Cholesky decomposition of H
l, denoted as H
l
=
R
l
T
R
l
,
(
l
=
1,2
).Proof.
Taking an inner product of (27) with u
n
+
1
+
u
n
-
1, from Lemma 5, we obtain (31)
1
2
τ
(
u
n
+
1
h
2
-
u
n
-
1
h
2
)
+
θ
(
H
2
δ
x
u
n
,
2
u
-
n
)
h
+
β
2
τ
(
R
1
δ
x
+
u
n
+
1
h
2
-
R
1
δ
x
+
u
n
-
1
h
2
)
=
0
.Noting that (32)
H
2
δ
x
u
n
,
u
n
+
1
+
u
n
-
1
h
=
(
R
2
δ
x
u
n
,
R
2
u
n
+
1
)
h
-
R
2
δ
x
u
n
-
1
,
R
2
u
n
h
,from (31)-(32), we obtain (33)
1
2
τ
(
u
n
+
1
h
2
-
u
n
-
1
h
2
)
+
θ
[
R
2
δ
x
u
n
,
R
2
u
n
+
1
h
-
R
2
δ
x
u
n
-
1
,
R
2
u
n
h
]
+
β
2
τ
(
R
1
δ
x
+
u
n
+
1
h
2
-
R
1
δ
x
+
u
n
-
1
h
2
)
=
0
.Let (34)
E
n
=
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
+
τ
θ
(
R
2
δ
x
u
n
,
R
2
u
n
+
1
)
.Then, from (33), we get E
n
=
E
n
-
1. This completes the proof.


### 3.2. Existence and Prior Estimates of Difference Solution

Theorem 10.
Suppose thatu
0
∈
H
0
1
(
Ω
); then the finite difference scheme (25) is solvable.Proof.
Assume that there existu
0
,
u
1
,
…
,
u
n which satisfy (25) as n
≤
N
-
1; now we try to prove that u
n
+
1 satisfy (25). We define the mapping g
:
R
0
J
→
R
0
J as follows: (35)
g
v
=
v
-
u
n
-
β
(
H
1
δ
x
+
δ
x
-
v
-
H
1
δ
x
+
δ
x
-
u
n
)
+
τ
H
2
δ
x
v
+
τ
φ
(
v
,
v
)is obviously continuous. Taking in (35) the inner product with v, from Lemmas 2, 4, and 5, we obtain(36)
g
v
,
v
h
=
v
h
2
-
(
v
,
u
n
)
h
+
β
[
R
1
δ
x
+
v
h
2
-
R
1
δ
x
+
v
,
R
1
δ
x
+
u
n
h
]
≥
1
2
(
v
h
2
-
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
v
h
2
-
R
1
δ
x
+
u
n
h
2
)
≥
1
2
(
v
h
2
-
u
n
h
2
)
-
3
β
4
δ
x
+
u
n
h
2
.
Thus forv
h
2
=
u
n
h
2
+
(
3
β
/
2
)
δ
x
+
u
n
h
2
+
1, there exists (
g
(
v
)
,
v
)
h
>
0. The existence of u
-
n follows from Lemma 7 and consequently the existence of u
n
+
1
=
2
v
-
u
n
-
1 is obtained. This completes the proof.Next we will give some a priori estimates of difference solutions.Lemma 11.
Suppose thatu
0
∈
H
0
1
(
Ω
); then there exists the estimation for the solution of problem (1)–(3): (37)
u
L
2
≤
C
,
u
x
L
2
≤
C
,
u
L
∞
≤
K
0
.Proof.
It follows from (5) that (38)
u
L
2
≤
C
,
u
x
L
2
≤
C
.Hence, it follows from the Sobolev inequality that (39)
u
L
∞
≤
K
0
.Lemma 12.
Suppose thatu
0
∈
H
0
1
(
Ω
); then there exists the estimation for the solution of the difference scheme (25): (40)
u
n
h
≤
K
1
,
δ
x
+
u
n
h
≤
K
2
,
u
n
∞
,
h
≤
K
3
,where K
4
=
max
⁡
{
6
E
0
/
(
3
-
5
θ
τ
)
,
2
E
0
},  K
2
=
max
⁡
{
6
E
0
/
(
3
β
-
5
θ
τ
)
,
2
E
0
/
β
}
,
K
3
=
ε
K
2
+
K
(
ε
,
n
)
K
1.Proof.
From Theorem9, we obtain (41)
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
≤
E
0
+
τ
θ
R
2
δ
x
u
n
,
R
2
u
n
+
1
h
≤
E
0
+
τ
θ
2
(
R
2
δ
x
u
n
h
2
+
R
2
u
n
+
1
h
2
)
;then from Lemma 4, we have (42)
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
δ
x
+
u
n
+
1
h
2
+
δ
x
+
u
n
h
2
)
≤
E
0
+
3
τ
θ
2
(
δ
x
+
u
n
h
2
+
u
n
+
1
h
2
)
.That is (43)
(
1
2
-
3
τ
θ
2
)
u
n
+
1
h
2
+
1
2
u
n
h
2
+
β
2
δ
x
+
u
n
+
1
2
+
(
β
2
-
3
τ
θ
2
)
δ
x
+
u
n
2
≤
E
0
;let τ be small, such that min
⁡
{
1
/
2
-
3
τ
θ
/
2
,
β
/
2
-
3
τ
θ
/
2
}
>
0; then we can get (44)
u
n
h
≤
K
1
,
δ
x
+
u
n
h
≤
K
2
,where (45)
K
1
=
max
⁡
⁡
2
E
0
1
-
3
θ
τ
,
2
E
0
,
K
2
=
max
⁡
⁡
2
E
0
β
-
3
θ
τ
,
2
E
0
β
.It follows from Lemma 6 that (46)
u
n
∞
,
h
≤
K
3
,where K
3
=
ε
K
2
+
K
(
ε
,
n
)
K
1.
This completes the proof.


### 3.3. Convergence and Stability of Difference Solution

First, we consider the truncation error of the finite difference scheme (25). Suppose that v
j
n
=
u
(
x
j
,
t
n
), which is the solution of problem (1)–(3). Then we have (47)
r
n
=
∂
t
v
n
+
θ
H
2
δ
x
v
n
+
(
1
-
θ
)
H
2
δ
x
v
-
n
-
β
H
1
δ
x
+
δ
x
-
∂
t
v
n
+
p
α
p
+
1
[
v
-
n
p
-
1
H
2
δ
x
v
-
n
+
H
2
δ
x
v
-
n
p
]
;according to Taylor’s expansion, r
j
n
=
O
(
τ
2
+
h
4
) can be easily obtained. Next, we consider convergence and stability of the finite difference scheme (25).Theorem 13.
Suppose thatu
0
∈
H
0
1
(
Ω
) and u
∈
C
(
5,3
); then the solution of the conservative difference scheme (25) converges to the solution of problem (1)–(3) with the order O
(
τ
2
+
h
4
) by L
∞ norm.Proof.
Subtracting (27) from (47), and letting e
n
=
v
n
-
u
n, we have (48)
r
n
=
∂
t
e
n
+
θ
H
2
δ
x
e
n
+
(
1
-
θ
)
H
2
δ
x
e
-
n
-
β
H
1
δ
x
+
δ
x
-
∂
t
e
n
+
φ
(
v
-
n
,
v
-
n
)
-
φ
(
u
-
n
,
u
-
n
)
;taking an inner product of (48) with e
n
+
1
+
e
n
-
1, we obtain (49)
r
n
,
e
n
+
1
+
e
n
-
1
h
=
1
2
τ
(
e
n
+
1
h
2
-
e
n
-
1
h
2
)
+
β
2
τ
(
R
1
δ
x
+
e
n
+
1
h
2
-
R
1
δ
x
+
e
x
n
-
1
h
2
)
+
θ
R
2
δ
x
e
n
,
e
n
+
1
+
e
n
-
1
h
+
φ
v
-
n
,
v
-
n
-
φ
u
-
n
,
u
-
n
,
e
n
+
1
+
e
n
-
1
h
;from Lemma 4 and Cauchy-Schwarz inequality, we obtain (50)
R
2
δ
x
e
n
,
e
n
+
1
+
e
n
-
1
h
≤
3
δ
x
e
x
n
h
2
+
1
2
e
n
+
1
h
2
+
1
2
e
n
-
1
h
2
;according to Lemmas 2 and 4, we have (51)
φ
v
-
n
,
v
-
n
-
φ
u
-
n
,
u
-
n
,
e
n
+
1
+
e
n
-
1
h
=
2
p
α
p
+
1
h
∑
j
=
1
J
-
1
‍
[
v
-
j
n
p
-
1
A
2
-
1
δ
x
v
-
j
n
-
u
-
j
n
p
-
1
A
2
-
1
δ
x
u
-
j
n
]
e
-
j
n
=
2
p
α
p
+
1
ll
k
+
h
∑
j
=
0
J
-
1
‍
[
A
2
-
1
δ
x
v
-
j
n
p
-
A
2
-
1
δ
x
u
-
j
n
p
]
e
-
j
n
≤
K
4
h
∑
j
=
0
J
-
1
‍
A
2
-
1
δ
x
e
-
j
n
+
e
-
j
n
e
-
j
n
+
K
4
h
∑
j
=
0
J
-
1
‍
e
-
j
n
A
2
-
1
δ
x
e
-
j
n
≤
K
4
(
9
δ
x
e
-
n
h
2
+
2
e
-
n
h
2
)
≤
K
4
9
2
δ
x
e
n
+
1
h
2
+
9
2
δ
x
e
n
-
1
h
2
+
e
n
+
1
h
2
+
e
n
-
1
h
2
,where K
4
=
(
p
α
/
2
p
+
1
)
max
⁡
{
K
0
p
-
1
,
2
(
p
-
1
)
K
0
p
-
2
K
3
,
2
(
p
-
1
)
K
3
p
-
1
}.
Substituting (50) and (51) into (49), we obtain (52)
1
2
τ
e
n
+
1
h
2
-
e
n
-
1
h
2
+
β
2
τ
(
R
1
δ
x
+
e
n
+
1
h
2
-
R
1
δ
x
+
e
x
n
-
1
h
2
)
≤
r
n
h
2
+
1
2
(
e
n
+
1
h
2
+
e
n
-
1
h
2
)
+
θ
(
3
δ
x
e
x
n
h
2
+
1
2
e
n
+
1
h
2
+
1
2
e
n
-
1
h
2
)
+
K
4
(
9
2
δ
x
e
n
+
1
h
2
+
9
2
δ
x
e
n
-
1
h
2
+
e
n
+
1
h
2
+
e
n
-
1
h
2
)
,that is, (53)
1
2
(
e
n
+
1
h
2
-
e
n
-
1
h
2
)
+
β
2
(
δ
x
+
e
n
+
1
h
2
-
δ
x
+
e
x
n
-
1
h
2
)
≤
τ
K
5
e
n
+
1
h
2
+
e
n
h
2
+
e
n
-
1
h
2
+
δ
x
+
e
n
+
1
h
2
+
δ
x
+
e
n
h
2
+
δ
x
+
e
n
-
1
h
2
+
τ
r
n
h
2
,where K
5
=
max
⁡
{
1
+
θ
/
2
+
K
4
,
(
9
/
2
)
K
4
,
3
θ
}. Let (54)
B
n
=
1
2
(
e
n
+
1
h
2
+
e
n
h
2
)
+
β
2
(
δ
x
+
e
n
+
1
h
2
+
δ
x
+
e
n
h
2
)
;then (53) can be rewritten as (55)
B
n
-
B
n
-
1
≤
τ
r
n
h
2
+
τ
K
6
(
B
n
+
B
n
-
1
)
,where K
6
=
max
⁡
{
2
K
5
,
2
K
5
/
β
}. From Lemma 8, we have (56)
B
N
≤
(
B
0
+
T
sup
⁡
1
≤
n
≤
N
r
n
h
2
)
e
4
K
6
T
.Thus we can choose a fourth-order method to compute u
1 such that (57)
B
0
≤
[
O
(
τ
2
+
h
4
)
]
2
;it follows from (56) that (58)
e
n
h
≤
O
(
τ
2
+
h
4
)
,
δ
x
+
e
n
h
≤
O
(
τ
2
+
h
4
)
,and then, from Lemma 6, we obtain (59)
e
∞
,
h
≤
O
(
τ
2
+
h
4
)
.This completes the proof.Below, we can similarly prove stability of the difference solution.Theorem 14.
Under the conditions of Theorem13, the solution of conservative finite difference scheme (25) is stable by L
∞ norm.


## 3.1. The Nonlinear-Implicit Scheme and Its Conservative Law

Next we consider the compact finite difference scheme for problem (1)–(3) as follows: (25)
∂
t
u
j
n
+
θ
A
2
-
1
δ
x
u
j
n
+
(
1
-
θ
)
A
2
-
1
δ
x
u
-
j
n
+
p
α
p
+
1
[
u
-
j
n
p
-
1
A
2
-
1
δ
x
u
-
j
n
+
A
2
-
1
δ
x
u
-
j
n
p
]
-
β
A
1
-
1
δ
x
+
δ
x
-
(
∂
t
u
j
n
)
=
0
,
1
≤
j
≤
J
-
1
,
1
≤
n
≤
N
-
1
,
u
0
n
=
u
J
n
=
0
,
1
≤
n
≤
N
,
u
j
0
=
u
0
(
x
j
)
,
0
≤
j
≤
J
,where weight coefficient θ
∈
[
0,1
]. Note that we need another finite difference scheme to calculate u
1, so the following scheme will be used:(26)
u
j
1
-
β
A
1
-
1
δ
x
+
δ
x
-
u
j
1
=
u
0
(
x
j
)
-
β
∂
2
u
0
∂
x
2
(
x
j
)
-
τ
∂
u
0
∂
x
(
x
j
)
-
τ
u
0
(
x
j
)
∂
u
0
∂
x
(
x
j
)
.The matrix form of the difference scheme (25) can be written as(27)
∂
t
u
n
+
θ
H
2
δ
x
u
n
+
(
1
-
θ
)
H
2
δ
x
u
-
n
+
p
α
p
+
1
[
u
-
n
p
-
1
H
2
δ
x
u
-
n
+
H
2
δ
x
u
-
n
p
]
-
β
H
1
δ
x
+
δ
x
-
(
∂
t
u
n
)
=
0
,
1
≤
n
≤
N
-
1
,
(28)
u
0
n
=
u
J
n
=
0
,
1
≤
n
≤
N
,
(29)
u
j
0
=
u
0
(
x
j
)
,
0
≤
j
≤
J
.Theorem 9.
Suppose thatu
0
∈
H
0
1
(
Ω
); then the finite difference scheme (25) is conservative for discrete energy; that is, (30)
E
n
=
1
2
u
n
+
1
h
2
+
u
n
h
2
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
+
τ
θ
R
2
δ
x
u
n
,
R
2
u
n
+
1
=
E
n
-
1
=
⋯
=
E
0
,where R
l are obtained by Cholesky decomposition of H
l, denoted as H
l
=
R
l
T
R
l
,
(
l
=
1,2
).Proof.
Taking an inner product of (27) with u
n
+
1
+
u
n
-
1, from Lemma 5, we obtain (31)
1
2
τ
(
u
n
+
1
h
2
-
u
n
-
1
h
2
)
+
θ
(
H
2
δ
x
u
n
,
2
u
-
n
)
h
+
β
2
τ
(
R
1
δ
x
+
u
n
+
1
h
2
-
R
1
δ
x
+
u
n
-
1
h
2
)
=
0
.Noting that (32)
H
2
δ
x
u
n
,
u
n
+
1
+
u
n
-
1
h
=
(
R
2
δ
x
u
n
,
R
2
u
n
+
1
)
h
-
R
2
δ
x
u
n
-
1
,
R
2
u
n
h
,from (31)-(32), we obtain (33)
1
2
τ
(
u
n
+
1
h
2
-
u
n
-
1
h
2
)
+
θ
[
R
2
δ
x
u
n
,
R
2
u
n
+
1
h
-
R
2
δ
x
u
n
-
1
,
R
2
u
n
h
]
+
β
2
τ
(
R
1
δ
x
+
u
n
+
1
h
2
-
R
1
δ
x
+
u
n
-
1
h
2
)
=
0
.Let (34)
E
n
=
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
+
τ
θ
(
R
2
δ
x
u
n
,
R
2
u
n
+
1
)
.Then, from (33), we get E
n
=
E
n
-
1. This completes the proof.


## 3.2. Existence and Prior Estimates of Difference Solution

Theorem 10.
Suppose thatu
0
∈
H
0
1
(
Ω
); then the finite difference scheme (25) is solvable.Proof.
Assume that there existu
0
,
u
1
,
…
,
u
n which satisfy (25) as n
≤
N
-
1; now we try to prove that u
n
+
1 satisfy (25). We define the mapping g
:
R
0
J
→
R
0
J as follows: (35)
g
v
=
v
-
u
n
-
β
(
H
1
δ
x
+
δ
x
-
v
-
H
1
δ
x
+
δ
x
-
u
n
)
+
τ
H
2
δ
x
v
+
τ
φ
(
v
,
v
)is obviously continuous. Taking in (35) the inner product with v, from Lemmas 2, 4, and 5, we obtain(36)
g
v
,
v
h
=
v
h
2
-
(
v
,
u
n
)
h
+
β
[
R
1
δ
x
+
v
h
2
-
R
1
δ
x
+
v
,
R
1
δ
x
+
u
n
h
]
≥
1
2
(
v
h
2
-
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
v
h
2
-
R
1
δ
x
+
u
n
h
2
)
≥
1
2
(
v
h
2
-
u
n
h
2
)
-
3
β
4
δ
x
+
u
n
h
2
.
Thus forv
h
2
=
u
n
h
2
+
(
3
β
/
2
)
δ
x
+
u
n
h
2
+
1, there exists (
g
(
v
)
,
v
)
h
>
0. The existence of u
-
n follows from Lemma 7 and consequently the existence of u
n
+
1
=
2
v
-
u
n
-
1 is obtained. This completes the proof.Next we will give some a priori estimates of difference solutions.Lemma 11.
Suppose thatu
0
∈
H
0
1
(
Ω
); then there exists the estimation for the solution of problem (1)–(3): (37)
u
L
2
≤
C
,
u
x
L
2
≤
C
,
u
L
∞
≤
K
0
.Proof.
It follows from (5) that (38)
u
L
2
≤
C
,
u
x
L
2
≤
C
.Hence, it follows from the Sobolev inequality that (39)
u
L
∞
≤
K
0
.Lemma 12.
Suppose thatu
0
∈
H
0
1
(
Ω
); then there exists the estimation for the solution of the difference scheme (25): (40)
u
n
h
≤
K
1
,
δ
x
+
u
n
h
≤
K
2
,
u
n
∞
,
h
≤
K
3
,where K
4
=
max
⁡
{
6
E
0
/
(
3
-
5
θ
τ
)
,
2
E
0
},  K
2
=
max
⁡
{
6
E
0
/
(
3
β
-
5
θ
τ
)
,
2
E
0
/
β
}
,
K
3
=
ε
K
2
+
K
(
ε
,
n
)
K
1.Proof.
From Theorem9, we obtain (41)
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
R
1
δ
x
+
u
n
+
1
h
2
+
R
1
δ
x
+
u
n
h
2
)
≤
E
0
+
τ
θ
R
2
δ
x
u
n
,
R
2
u
n
+
1
h
≤
E
0
+
τ
θ
2
(
R
2
δ
x
u
n
h
2
+
R
2
u
n
+
1
h
2
)
;then from Lemma 4, we have (42)
1
2
(
u
n
+
1
h
2
+
u
n
h
2
)
+
β
2
(
δ
x
+
u
n
+
1
h
2
+
δ
x
+
u
n
h
2
)
≤
E
0
+
3
τ
θ
2
(
δ
x
+
u
n
h
2
+
u
n
+
1
h
2
)
.That is (43)
(
1
2
-
3
τ
θ
2
)
u
n
+
1
h
2
+
1
2
u
n
h
2
+
β
2
δ
x
+
u
n
+
1
2
+
(
β
2
-
3
τ
θ
2
)
δ
x
+
u
n
2
≤
E
0
;let τ be small, such that min
⁡
{
1
/
2
-
3
τ
θ
/
2
,
β
/
2
-
3
τ
θ
/
2
}
>
0; then we can get (44)
u
n
h
≤
K
1
,
δ
x
+
u
n
h
≤
K
2
,where (45)
K
1
=
max
⁡
⁡
2
E
0
1
-
3
θ
τ
,
2
E
0
,
K
2
=
max
⁡
⁡
2
E
0
β
-
3
θ
τ
,
2
E
0
β
.It follows from Lemma 6 that (46)
u
n
∞
,
h
≤
K
3
,where K
3
=
ε
K
2
+
K
(
ε
,
n
)
K
1.
This completes the proof.


## 3.3. Convergence and Stability of Difference Solution

First, we consider the truncation error of the finite difference scheme (25). Suppose that v
j
n
=
u
(
x
j
,
t
n
), which is the solution of problem (1)–(3). Then we have (47)
r
n
=
∂
t
v
n
+
θ
H
2
δ
x
v
n
+
(
1
-
θ
)
H
2
δ
x
v
-
n
-
β
H
1
δ
x
+
δ
x
-
∂
t
v
n
+
p
α
p
+
1
[
v
-
n
p
-
1
H
2
δ
x
v
-
n
+
H
2
δ
x
v
-
n
p
]
;according to Taylor’s expansion, r
j
n
=
O
(
τ
2
+
h
4
) can be easily obtained. Next, we consider convergence and stability of the finite difference scheme (25).Theorem 13.
Suppose thatu
0
∈
H
0
1
(
Ω
) and u
∈
C
(
5,3
); then the solution of the conservative difference scheme (25) converges to the solution of problem (1)–(3) with the order O
(
τ
2
+
h
4
) by L
∞ norm.Proof.
Subtracting (27) from (47), and letting e
n
=
v
n
-
u
n, we have (48)
r
n
=
∂
t
e
n
+
θ
H
2
δ
x
e
n
+
(
1
-
θ
)
H
2
δ
x
e
-
n
-
β
H
1
δ
x
+
δ
x
-
∂
t
e
n
+
φ
(
v
-
n
,
v
-
n
)
-
φ
(
u
-
n
,
u
-
n
)
;taking an inner product of (48) with e
n
+
1
+
e
n
-
1, we obtain (49)
r
n
,
e
n
+
1
+
e
n
-
1
h
=
1
2
τ
(
e
n
+
1
h
2
-
e
n
-
1
h
2
)
+
β
2
τ
(
R
1
δ
x
+
e
n
+
1
h
2
-
R
1
δ
x
+
e
x
n
-
1
h
2
)
+
θ
R
2
δ
x
e
n
,
e
n
+
1
+
e
n
-
1
h
+
φ
v
-
n
,
v
-
n
-
φ
u
-
n
,
u
-
n
,
e
n
+
1
+
e
n
-
1
h
;from Lemma 4 and Cauchy-Schwarz inequality, we obtain (50)
R
2
δ
x
e
n
,
e
n
+
1
+
e
n
-
1
h
≤
3
δ
x
e
x
n
h
2
+
1
2
e
n
+
1
h
2
+
1
2
e
n
-
1
h
2
;according to Lemmas 2 and 4, we have (51)
φ
v
-
n
,
v
-
n
-
φ
u
-
n
,
u
-
n
,
e
n
+
1
+
e
n
-
1
h
=
2
p
α
p
+
1
h
∑
j
=
1
J
-
1
‍
[
v
-
j
n
p
-
1
A
2
-
1
δ
x
v
-
j
n
-
u
-
j
n
p
-
1
A
2
-
1
δ
x
u
-
j
n
]
e
-
j
n
=
2
p
α
p
+
1
ll
k
+
h
∑
j
=
0
J
-
1
‍
[
A
2
-
1
δ
x
v
-
j
n
p
-
A
2
-
1
δ
x
u
-
j
n
p
]
e
-
j
n
≤
K
4
h
∑
j
=
0
J
-
1
‍
A
2
-
1
δ
x
e
-
j
n
+
e
-
j
n
e
-
j
n
+
K
4
h
∑
j
=
0
J
-
1
‍
e
-
j
n
A
2
-
1
δ
x
e
-
j
n
≤
K
4
(
9
δ
x
e
-
n
h
2
+
2
e
-
n
h
2
)
≤
K
4
9
2
δ
x
e
n
+
1
h
2
+
9
2
δ
x
e
n
-
1
h
2
+
e
n
+
1
h
2
+
e
n
-
1
h
2
,where K
4
=
(
p
α
/
2
p
+
1
)
max
⁡
{
K
0
p
-
1
,
2
(
p
-
1
)
K
0
p
-
2
K
3
,
2
(
p
-
1
)
K
3
p
-
1
}.
Substituting (50) and (51) into (49), we obtain (52)
1
2
τ
e
n
+
1
h
2
-
e
n
-
1
h
2
+
β
2
τ
(
R
1
δ
x
+
e
n
+
1
h
2
-
R
1
δ
x
+
e
x
n
-
1
h
2
)
≤
r
n
h
2
+
1
2
(
e
n
+
1
h
2
+
e
n
-
1
h
2
)
+
θ
(
3
δ
x
e
x
n
h
2
+
1
2
e
n
+
1
h
2
+
1
2
e
n
-
1
h
2
)
+
K
4
(
9
2
δ
x
e
n
+
1
h
2
+
9
2
δ
x
e
n
-
1
h
2
+
e
n
+
1
h
2
+
e
n
-
1
h
2
)
,that is, (53)
1
2
(
e
n
+
1
h
2
-
e
n
-
1
h
2
)
+
β
2
(
δ
x
+
e
n
+
1
h
2
-
δ
x
+
e
x
n
-
1
h
2
)
≤
τ
K
5
e
n
+
1
h
2
+
e
n
h
2
+
e
n
-
1
h
2
+
δ
x
+
e
n
+
1
h
2
+
δ
x
+
e
n
h
2
+
δ
x
+
e
n
-
1
h
2
+
τ
r
n
h
2
,where K
5
=
max
⁡
{
1
+
θ
/
2
+
K
4
,
(
9
/
2
)
K
4
,
3
θ
}. Let (54)
B
n
=
1
2
(
e
n
+
1
h
2
+
e
n
h
2
)
+
β
2
(
δ
x
+
e
n
+
1
h
2
+
δ
x
+
e
n
h
2
)
;then (53) can be rewritten as (55)
B
n
-
B
n
-
1
≤
τ
r
n
h
2
+
τ
K
6
(
B
n
+
B
n
-
1
)
,where K
6
=
max
⁡
{
2
K
5
,
2
K
5
/
β
}. From Lemma 8, we have (56)
B
N
≤
(
B
0
+
T
sup
⁡
1
≤
n
≤
N
r
n
h
2
)
e
4
K
6
T
.Thus we can choose a fourth-order method to compute u
1 such that (57)
B
0
≤
[
O
(
τ
2
+
h
4
)
]
2
;it follows from (56) that (58)
e
n
h
≤
O
(
τ
2
+
h
4
)
,
δ
x
+
e
n
h
≤
O
(
τ
2
+
h
4
)
,and then, from Lemma 6, we obtain (59)
e
∞
,
h
≤
O
(
τ
2
+
h
4
)
.This completes the proof.Below, we can similarly prove stability of the difference solution.Theorem 14.
Under the conditions of Theorem13, the solution of conservative finite difference scheme (25) is stable by L
∞ norm.


## 4. Numerical Experiments

In this section, two examples are presented to illustrate the effectiveness of the finite difference scheme (25) in [
-
50,50
]. The single solitary wave solution of (1) is (60)
u
(
x
,
t
)
=
A
·
sech
⁡
2
/
(
p
-
1
)
⁡
[
k
(
x
+
x
0
-
σ
t
)
]
,where (61)
A
=
p
+
1
σ
-
1
2
α
1
/
(
p
-
1
)
,
k
=
p
-
1
2
β
σ
-
1
σ
,and σ, x
0 are arbitrary constants and p
≥
2.Letx
0
=
0 in (60), α
=
1
/
2, β
=
1, and u
0
(
x
)
=
A
·
sech
⁡
2
/
(
p
-
1
)
(
k
x
) and consider two cases: p
=
2 and p
=
4. We take h
=
0.1 and τ
=
0.1, respectively. The errors e
n
∞
,
h are listed in Tables 1-2, respectively.Table 1
Errors computed by the proposed compact scheme withσ
=
4
/
3, p
=
2 and h
=
τ
=
0.1.
t
e
n
∞
,
h
θ
=
0
θ
=
0.25
θ
=
0.5
θ
=
0.75
θ
=
1
0.2
2.6213
×
1
0
-
5
1.2528
×
1
0
-
5
1.5794
×
1
0
-
5
3.2838
×
1
0
-
5
5.0401
×
1
0
-
5
0.4
5.2311
×
1
0
-
5
2.4917
×
1
0
-
5
3.1520
×
1
0
-
5
6.5530
×
1
0
-
5
1.0012
×
1
0
-
4
0.6
7.8509
×
1
0
-
5
3.7267
×
1
0
-
5
4.7095
×
1
0
-
5
9.7948
×
1
0
-
5
1.4917
×
1
0
-
4
0.8
1.0476
×
1
0
-
4
4.9503
×
1
0
-
5
6.2474
×
1
0
-
5
1.3006
×
1
0
-
4
1.9758
×
1
0
-
4
1.0
1.3090
×
1
0
-
4
6.1653
×
1
0
-
5
7.7647
×
1
0
-
5
1.6222
×
1
0
-
4
2.4638
×
1
0
-
4Table 2
Errors computed by the proposed compact scheme withσ
=
4
/
3, p
=
4 and h
=
τ
=
0.1.
t
e
n
∞
,
h
θ
=
0
θ
=
0.25
θ
=
0.5
θ
=
0.75
θ
=
1
0.2
1.2498
×
1
0
-
4
1.7684
×
1
0
-
4
2.3183
×
1
0
-
4
2.8729
×
1
0
-
4
3.4464
×
1
0
-
4
0.4
2.3629
×
1
0
-
4
3.4550
×
1
0
-
4
4.5445
×
1
0
-
4
5.6499
×
1
0
-
4
6.7531
×
1
0
-
4
0.6
3.3108
×
1
0
-
4
5.0029
×
1
0
-
4
6.6455
×
1
0
-
4
8.3064
×
1
0
-
4
9.9436
×
1
0
-
4
0.8
4.1205
×
1
0
-
4
6.4180
×
1
0
-
4
8.6651
×
1
0
-
4
1.0913
×
1
0
-
3
1.3159
×
1
0
-
3
1.0
4.9896
×
1
0
-
4
7.7376
×
1
0
-
4
1.0667
×
1
0
-
3
1.3595
×
1
0
-
3
1.6550
×
1
0
-
3In Table3, 4, and 5, the comparison of e
∞
,
h by the compact scheme for h
=
τ
=
0.1 with the Zhang [4] scheme for h
=
0.05, τ
=
0.1 when σ
=
4
/
3, θ
=
0.25 is shown. From Table 3, we can see that our compact scheme is acceptable.Table 3
Comparison ofe
∞
,
h by the compact scheme for h
=
τ
=
0.1 with the Zhang [4] scheme for h
=
0.05, τ
=
0.1.
t
p
=
2
p
=
4
Compact Scheme
Zhang [4]
Compact Scheme
Zhang [4]
0.2
1.2528
×
1
0
-
5
7.756
×
1
0
-
6
1.2498
×
1
0
-
4
0.0049
0.4
2.4917
×
1
0
-
5
1.575
×
1
0
-
5
2.3629
×
1
0
-
4
0.0098
0.6
3.7267
×
1
0
-
5
2.357
×
1
0
-
5
3.3108
×
1
0
-
4
0.0148
0.8
4.9503
×
1
0
-
5
3.129
×
1
0
-
5
4.1205
×
1
0
-
4
0.0198
1.0
6.1653
×
1
0
-
5
3.963
×
1
0
-
5
4.9896
×
1
0
-
4
0.0239Table 4
Comparison ofe
∞
,
h by the compact scheme for σ
=
4
/
3, θ
=
0.25, p
=
2, τ
=
0.1 and h
=
0.05.
t
e
n
∞
,
h
Compact Scheme
Shao et al. [7]
C-N scheme
Raslan [10]
Dogan [11]
0.2
1.2536
×
1
0
-
5
0.00056
0.00070
0.00190
0.00053
0.4
2.4916
×
1
0
-
5
0.00085
0.03331
0.00283
0.00113
0.6
3.7265
×
1
0
-
5
0.00112
0.06337
0.00403
0.00175
0.8
4.9526
×
1
0
-
5
0.00141
0.08433
0.00481
0.00237
1.0
6.1667
×
1
0
-
5
0.00169
0.11287
0.00563
0.00299Table 5
Comparison ofe
∞
,
h by the compact scheme for σ
=
1.03, θ
=
0.25, p
=
2, τ
=
h
=
0.1.
t
e
n
∞
,
h
Compact Scheme
Bakodah and Banaja[6]
Kutluay and Esen[8]
4
7.0938
×
1
0
-
5
1.4805
×
1
0
-
4
1.23
×
1
0
-
4
8
2.1660
×
1
0
-
4
2.9961
×
1
0
-
4
1.66
×
1
0
-
4
12
5.1012
×
1
0
-
4
4.5367
×
1
0
-
4
1.79
×
1
0
-
4Numerical results show that numerical precision depends on the choice of parameterθ. From Tables 1-2, e
n
∞
,
h
≤
O
(
τ
2
+
h
4
) is validated. We take different p, h, and θ values and compute the errors for the solution of problem (1)–(3). Numerical results are almost identical with the above experiment result. Hence, our schemes are efficient and reliable.In Figures1, 2, 3, 4, 5, and 6, we show the numerical solution and conservative discrete energy in each case.Figure 1
Numerical solutionu
n of scheme with p
=
2 and τ
=
h
=
0.1.Figure 2
Discrete energyE
n of scheme with p
=
2, T
=
10, and τ
=
h
=
0.1.Figure 3
Absolute errore
N of scheme with p
=
2 and τ
=
h
=
0.1.Figure 4
Numerical solutionu
n of scheme with p
=
4 and τ
=
h
=
0.1.Figure 5
Discrete energyE
n of scheme with p
=
4, T
=
10, and τ
=
h
=
0.1.Figure 6
Absolute errore
N of scheme with p
=
4 and τ
=
h
=
0.1.



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*Source: 960602-2015-10-25.xml*