A new conservative finite difference scheme is presented for an initial-boundary value problem of the general Rosenau-RLW equation. Existence of its difference solutions are proved by Brouwer fixed point theorem. It is proved by the discrete energy method that the scheme is uniquely solvable, unconditionally stable, and second-order convergent. Numerical examples show the efficiency of the scheme.
1. Introduction
In this paper, we consider the following initial-boundary value problem of the general Rosenau-RLW equation:
(11)
with an initial condition
(12)
and boundary conditions
(13)
where is a integer and is a known smooth function. When , (1.1) is called as usual Rosenau-RLW equation. When , (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:
(14)
(15)
It is known the conservative scheme is better than the nonconservative ones. Zhang et al. [1] point out that the nonconservative scheme may easily show nonlinear blow up. In [2] Li and Vu-Quoc said " in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation". In [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general Rosenau-RLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
2. A Nonlinear-Implicit Conservative Scheme
In this section, we propose a nonlinear-implicit conservative scheme for the initial-boundary value problem (1.1)–(1.3) and give its numerical analysis.
2.1. The Nonlinear-Implicit Scheme and Its Conservative Law
For convenience, we introduce the following notations
(21)
where and denote the spatial and temporal mesh sizes, , , respectively,
(22)
and in the paper, denotes a general positive constant, which may have different values in different occurrences.
Since , then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:
Suppose that , then scheme (2.3)–(2.5) is conservative in the senses:
(28)
(29)
Proof.
Multiplying (2.3) with , according to boundary condition (2.5), and then summing up for from 1 to , we have
(210)
Let
(211)
Then (2.8) is gotten from (2.10).
Computing the inner product of (2.3) with , according to boundary condition (2.5) and Lemma 2.1, we obtain
(212)
where
(213)
According to
(214)
we have . It follows from (2.12) that
(215)
Let
(216)
Then (2.9) is gotten from (2.15). This completes the proof of Theorem 2.2.
2.2. Existence and Prior Estimates of Difference Solution
To show the existence of the approximations for scheme (2.3)–(2.5), we introduce the following Brouwer fixed point theorem [13].
Lemma 2.3.
Let be a finite-dimensional inner product space, be the associated norm, and be continuous. Assume, moreover, that there exist , for all , , . Then, there exists a such that and .
Let , , then have the following.
Theorem 2.4.
There exists which satisfies scheme (2.3)–(2.5).
Proof.
It follows from the original problem (1.1)–(1.3) that satisfies scheme (2.3)–(2.5). Assume there exists which satisfy scheme (2.3)–(2.5), as , now we try to prove that , satisfy scheme (2.3)–(2.5).
We define on as follows:
(217)
where . Computing the inner product of (2.17) with and considering and , we obtain
(218)
Hence, for all , there exists . It follows from Lemma 2.3 that exists which satisfies . Let , then it can be proved that is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.
Next we will give some priori estimates of difference solutions. First the following two lemmas [14] are introduced:
Lemma 2.5 (discrete Sobolev's estimate).
For any discrete function on the finite interval , there is the inequality
(219)
where are two constants independent of and step length .
Lemma 2.6 (discrete Gronwall's inequality).
Suppose that the discrete function satisfies the inequality
(220)
where and are nonnegative constants. Then
(221)
where is sufficiently small, such that .
Theorem 2.7.
Suppose that , then the following inequalities
(222)
hold.
Proof.
It is follows from (2.9) that
(223)
According to Lemma 2.5, we obtain
(224)
This completes the proof of Theorem 2.7.
Remark 2.8.
Theorem 2.7 implies that scheme (2.3)–(2.5) is unconditionally stable.
2.3. Convergence and Uniqueness of Difference Solution
First, we consider the convergence of scheme (2.3)–(2.5). We define the truncation error as follows:
(225)
then from Taylor's expansion, we obtain the following.
Theorem 2.9.
Suppose that and , then the truncation errors of scheme (2.3)–(2.5) satisfy
(226)
as ,
Theorem 2.10.
Suppose that the conditions of Theorem 2.9 are satisfied, then the solution of scheme (2.3)–(2.5) converges to the solution of problem (1.1)–(1.3) with order in the norm.
Proof.
Subtracting (2.3) from (2.25) letting
(227)
we obtain
(228)
Computing the inner product of (2.28) with , we obtain
(229)
From the conservative property (1.5), it can be proved by Lemma 2.5 that . Then by Theorem 2.7 we can estimate (2.29) as follows:
Choosing suitable which is small enough, we obtain by Lemma 2.6 that
(235)
From the discrete initial conditions, we know that is of second-order accuracy, then
(236)
Then we have
(237)
It follows from Lemma 2.5, we have . This completes the proof of Theorem 2.10.
Theorem 2.11.
Scheme (2.3)–(2.5) is uniquely solvable.
Proof.
Assume that and both satisfy scheme (2.3)–(2.5), let , we obtain
(238)
Similarly to the proof of Theorem 2.10, we have
(239)
This completes the proof of Theorem 2.11.
Remark 2.12.
All results above in this paper are correct for initial-boundary value problem of the general Rosenau-RLW equation with finite or infinite boundary.
3. Numerical Experiments
In order to test the correction of the numerical analysis in this paper, we consider the following initial-boundary value problems of the general Rosenau-RLW equation:
(31)
with an initial condition
(32)
and boundary conditions
(33)
where . Then the exact solution of the initial value problem (3.1)-(3.2) is
(34)
where is wave velocity.
It follows from (3.4) that the initial-boundary value problem (3.1)–(3.3) is consistent to the boundary value problem (3.3) for . In the following examples, we always choose , .
Tables 1, 2, and 3 give the errors in the sense of -norm and -norm of the numerical solutions under various steps of and at for and . The three tables verify the second-order convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass and discrete energy computed by scheme (2.3)–(2.5) for and .
Table 1
The errors of numerical solutions at with for .
0.4
0.2
3.953 296
3.930 200
0.1
3.987 130
3.980 050
0.05
3.997 412
3.994 759
0.025
4.221 051
4.151 348
Table 2
The errors of numerical solutions at with for .
0.4
0.2
3.961 294
3.934 592
0.1
3.996 350
3.988 283
0.05
4.003 273
4.000 165
0.025
4.290 286
4.242 688
Table 3
The errors of numerical solutions at with for .
0.4
0.2
3.885 945
3.851 797
0.1
3.975 084
3.965 980
0.05
4.000 294
3.995 992
0.025
4.391 818
4.381 199
Table 4
Discrete mass and discrete energy with at various for .
10
1.897 658 262 960 01
0.533 175 231 580 85
20
1.897 658 268 873 21
0.533 175 231 872 51
30
1.897 658 262 993 93
0.533 175 231 177 25
40
1.897 658 265 568 93
0.533 175 231 478 09
50
1.897 658 260 975 87
0.533 175 231 776 18
60
1.897 658 265 384 88
0.533 175 231 074 05
Table 5
Discrete mass and discrete energy with at various for .
10
2.672 608 675 265 30
1.113 462 678 852 70
20
2.672 608 676 236 58
1.113 462 678 465 22
30
2.672 608 674 147 13
1.113 462 678 083 94
40
2.672 608 672 639 88
1.113 462 678 711 58
50
2.672 608 672 874 71
1.113 462 678 330 06
60
2.672 608 679 729 44
1.113 462 678 958 71
Table 6
Discrete mass and discrete energy with at various for .
10
3.988 663 320 390 89
1.917 613 014 656 71
20
3.988 663 260 854 26
1.917 613 014 739 89
30
3.988 663 167 685 49
1.917 613 014 820 83
40
3.988 663 194 506 97
1.917 613 014 927 44
50
3.988 663 973 359 89
1.917 613 014 009 71
60
3.988 663 621 972 59
1.917 613 014 679 17
Figures 1, 2, and 3 plot the exact solutions at and the numerical solutions computed by scheme (2.3)–(2.5) with at , which also show the accuracy of scheme (2.3)–(2.5).
Figure 1
Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .
Figure 2
Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .
Figure 3
Exact solutions of at and numerical solutions computed by scheme (2. 3)–(2.5) at for .
Declarations
Acknowledgments
The authors would like to express their sincere thanks to the referees for their valuable suggestions and comments. This paper is supported by the National Natural Science Foundation of China (nos. 10871117 and 10571110).
Authors’ Affiliations
(1)
School of Science, Shandong University of Technology
(2)
School of Mathematics, Shandong University
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is a integer and
is a known smooth function. When
, (1.1) is called as usual Rosenau-RLW equation. When
, (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:
is a integer and
is a known smooth function. When
, (1.1) is called as usual Rosenau-RLW equation. When
, (1.1) is called as modified Rosenau-RLW (MRosenau-RLW) equation. The initial boundary value problem (1.1)–(1.3) possesses the following conservative quantities:
in some areas, the ability to preserve some invariant properties of the original differential equation is a criterion to judge the success of a numerical simulation". In [3–11], some conservative finite difference schemes were used for a system of the generalized nonlinear Schrödinger equations, Regularized long wave (RLW) equations, Sine-Gordon equation, Klein-Gordon equation, Zakharov equations, Rosenau equation, respectively. Numerical results of all the schemes are very good. Hence, we propose a new conservative difference scheme for the general Rosenau-RLW equation, which simulates conservative laws (1.4) and (1.5) at the same time. The outline of the paper is as follows. In Section 2, a nonlinear difference scheme is proposed and corresponding convergence and stability of the scheme are proved. In Section 3, some numerical experiments are shown.
and
denote the spatial and temporal mesh sizes,
,
, respectively,
and
denote the spatial and temporal mesh sizes,
,
, respectively,
denotes a general positive constant, which may have different values in different occurrences.
, then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:
, then the finite difference scheme for the problem (1.1)–(1.3) is written as follows:
, one has
, one has
, then
, then
, then scheme (2.3)–(2.5) is conservative in the senses:
, then scheme (2.3)–(2.5) is conservative in the senses:
, according to boundary condition (2.5), and then summing up for
from 1 to
, we have
, according to boundary condition (2.5), and then summing up for
from 1 to
, we have
, according to boundary condition (2.5) and Lemma 2.1, we obtain
, according to boundary condition (2.5) and Lemma 2.1, we obtain
. It follows from (2.12) that
. It follows from (2.12) that
for scheme (2.3)–(2.5), we introduce the following Brouwer fixed point theorem [13].
be a finite-dimensional inner product space,
be the associated norm, and
be continuous. Assume, moreover, that there exist
, for all
,
,
. Then, there exists a
such that
and
.
,
, then have the following.
which satisfies scheme (2.3)–(2.5).
satisfies scheme (2.3)–(2.5). Assume there exists
which satisfy scheme (2.3)–(2.5), as
, now we try to prove that
, satisfy scheme (2.3)–(2.5).
on
as follows:
on
as follows:
. Computing the inner product of (2.17) with
and considering
and
, we obtain
. Computing the inner product of (2.17) with
and considering
and
, we obtain
,
there exists
. It follows from Lemma 2.3 that exists
which satisfies
. Let
, then it can be proved that
is the solution of scheme (2.3)–(2.5). This completes the proof of Theorem 2.4.
on the finite interval
, there is the inequality
on the finite interval
, there is the inequality
are two constants independent of
and step length
.
satisfies the inequality
satisfies the inequality
and
are nonnegative constants. Then
and
are nonnegative constants. Then
is sufficiently small, such that
.
, then the following inequalities
, then the following inequalities
and
, then the truncation errors of scheme (2.3)–(2.5) satisfy
and
, then the truncation errors of scheme (2.3)–(2.5) satisfy
,
in the
norm.
, we obtain
, we obtain
. Then by Theorem 2.7 we can estimate (2.29) as follows:
. Then by Theorem 2.7 we can estimate (2.29) as follows:
which is small enough, we obtain by Lemma 2.6 that
which is small enough, we obtain by Lemma 2.6 that
is of second-order accuracy, then
is of second-order accuracy, then
. This completes the proof of Theorem 2.10.
and
both satisfy scheme (2.3)–(2.5), let
, we obtain
and
both satisfy scheme (2.3)–(2.5), let
, we obtain
. Then the exact solution of the initial value problem (3.1)-(3.2) is
. Then the exact solution of the initial value problem (3.1)-(3.2) is
is wave velocity.
. In the following examples, we always choose
,
.
-norm and
-norm of the numerical solutions under various steps of
and
at
for
and
. The three tables verify the second-order convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass
and discrete energy
computed by scheme (2.3)–(2.5) for
and
.
-norm and
-norm of the numerical solutions under various steps of
and
at
for
and
. The three tables verify the second-order convergence and good stability of the numerical solutions. Tables 4, 5, and 6 shows the conservative law of discrete mass
and discrete energy
computed by scheme (2.3)–(2.5) for
and
.
with
for
.
with
for
.
with
for
.
and discrete energy
with
at various
for
.
and discrete energy
with
at various
for
.
and discrete energy
with
at various
for
.
and the numerical solutions computed by scheme (2.3)–(2.5) with
at
, which also show the accuracy of scheme (2.3)–(2.5).
and the numerical solutions computed by scheme (2.3)–(2.5) with
at
, which also show the accuracy of scheme (2.3)–(2.5).
at
and numerical solutions computed by scheme (2. 3)–(2.5) at
for
.
at
and numerical solutions computed by scheme (2. 3)–(2.5) at
for
.
at
and numerical solutions computed by scheme (2. 3)–(2.5) at
for
.