Numerical solutions for generalized Rosenau equation are considered and two energy conservative finite difference schemes are proposed. Existence of the solutions for the difference scheme has been shown. Stability, convergence, and priori error estimate of the scheme are proved using energy method. Numerical results demonstrate that two schemes are efficient and reliable.
1. Introduction
Consider the following initialboundary value problem for generalized Rosenau equation:
with an initial condition
and boundary conditions
where is a integer.
When , (1.1) is called as usual Rosenau equation proposed by Rosenau [1] for treating the dynamics of dense discrete systems. Since then, the Cauchy problem of the Rosenau equation was investigated by Park [2]. Many numerical schemes have been proposed, such as conforming finite element method by Chung and Pani [3], discontinuous Galerkin method by Choo et al. [4], orthogonal cubic spline collocation method by Manickam [5], and finite difference method by Chung [6] and Omrani et al. [7]. As for the generalized case, however, there are few studies on theoretical analysis and numerical methods.
It can be proved easily that the problem (1.1)–(1.3) has the following conservative law:
Hence, we propose two conservative difference schemes which simulate conservative law (1.4). 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, a linearized difference scheme is proposed and theoretical results are obtained. In Section 4, some numerical experiments are shown.
2. Nonlinear Finite Difference Scheme
Let and be the uniform step size in the spatial and temporal direction, respectively. Denote , , and . Define
and in the paper, denotes a general positive constant which may have different values in different occurrences.
Since , then the following finite difference scheme is considered:
Lemma 2.1 (see [8]).
For any two mesh functions, , one has
Furthermore, if , then
Theorem 2.2.
Suppose , then the scheme (2.2)–(2.4) is conservative for discrete energy, that is,
Proof.
Computing the inner product of (2.2) with , according to boundary condition (2.4) and Lemma 2.1, we have
where
According to
we obtain
By the definition of , (2.7) holds.
To prove the existence of solution for scheme (2.2)–(2.4), the following Browder fixed point Theorem should be introduced. For the proof, see [9].
Lemma 2.3 (Browder fixed point Theorem).
Let be a finite dimensional inner product space. Suppose that is continuous and there exists an such that for all with . Then there exists such that and .
Theorem 2.4.
There exists satisfying the difference scheme (2.2)–(2.4).
Proof.
By the mathematical induction, for , assume that satisfy (2.2)–(2.4). Next we prove that there exists satisfying (2.2)–(2.4).
Define a operator on as follows:
Taking the inner product of (2.13) with , we get
Obviously, for all , with . It follows from Lemma 2.3 that there exists which satisfies . Let , it can be proved that is the solution of the scheme (2.2)–(2.4).
Next, we discuss the convergence and stability of the scheme (2.2)–(2.4). Let be the solution of problem (1.1)–(1.3), , then the truncation of the scheme (2.2)–(2.4) is
Using Taylor expansion, we know that holds if .
Lemma 2.5.
Suppose that , then the solution of the initialboundary value problem (1.1)–(1.3) satisfies
Proof.
It follows from (1.4) that
Using Hölder inequality and Schwartz inequality, we get
Hence, . According to Sobolev inequality, we have .
Lemma 2.6 (Discrete Sobolev's inequality [10]).
There exist two constant and such that
Lemma 2.7 (Discrete Gronwall inequality [10]).
Suppose are nonnegative mesh functions and is nondecreasing. If and
then
Theorem 2.8.
Suppose , then the solution of (2.2) satisfies , which yield .
Proof.
It follows from (2.7) that
Using Lemma 2.1 and Schwartz inequality, we get
According to Lemma 2.6, we have .
Theorem 2.9.
Suppose , then the solution of the scheme (2.2)–(2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is .
Proof.
Subtracting (2.15) from (2.2) and letting , we have
Computing the inner product of (2.24) with , and using , we get
where
According to Lemma 2.5, Theorem 2.8, and Schwartz inequality, we have
Furthermore,
Substituting (2.27)–(2.29) into (2.25), we get
Similarly to the proof of (2.23), we have
and (2.30) can be rewritten as
Let , then (2.32) is written as follows:
If is sufficiently small which satisfies , then
Summing up (2.34) from to , we have
Noticing
and , we have . Hence
According to Lemma 2.7, we get , that is,
It follows from (2.31) that
By using Lemma 2.6, we have
This completes the proof of Theorem 2.9.
Similarly, the following theorem can be proved.
Theorem 2.10.
Under the conditions of Theorem 2.9, the solution of the scheme (2.2)–(2.4) is stable by .
3. Linearized Finite Difference Scheme
In this section, we propose a linearimplicit finite difference scheme as follows:
Theorem 3.1.
Suppose , then the scheme (3.1), (2.3), and (2.4) are conservative for discrete energy, that is,
Proof.
Computing the inner product of (3.1) with , we have
where
According to Lemma 2.1, we get
Adding (3.3) and (3.5) to (3.6), we obtain
By the definition of , (3.2) holds.
Theorem 3.2.
The difference scheme (3.1) is uniquely solvable.
Proof.
we use the mathematical induction. Obviously, is determined by (2.3) and we can choose a twoorder method to compute (e.g., by scheme (2.2)). Assuming that are uniquely solvable, consider in (3.1) which satisfies
Taking the inner product of (3.8) with , we get
Notice that
It follows from (3.8) that
That is, there uniquely exists trivial solution satisfying (3.8). Hence, in (3.1) is uniquely solvable.
To discuss the convergence and stability of the scheme (3.1), we denote the truncation of the scheme (3.1):
Using Taylor expansion, we know that holds if .
Theorem 3.3.
Suppose , then the solution of (3.1) satisfies , which yield .
Proof.
It follows from (3.2) that
According to (2.23), we have
that is,
If is sufficiently small which satisfies , we get
which yields . According to (2.23), we get
Using Lemma 2.6, we obtain
Theorem 3.4.
Suppose , then the solution of the schemes (3.1), (2.3), and (2.4) converges to the solution of problem (1.1)–(1.3) and the rate of convergence is .
Proof.
Subtracting (3.12) from (3.1) and letting , we have
Computing the inner product of (3.19) with , we get
where
Notice that
and similarly
Furthermore, we get
Substituting (3.22)–(3.24) into (3.20), we get
Similarly to the proof of (2.31), (3.25) can be written as
Let , then (3.26) is written as follows:
that is,
If is sufficiently small which satisfies , then
Summing up (3.29) from to , we have
Choosing a twoorder method to compute (e.g., by scheme (2.2)) and noticing
we have
According to Lemma 2.7, we get , that is,
According to (2.31), we get
By using Lemma 2.6, we have
This completes the proof of Theorem 3.4.
Similarly, the following theorem can be proved that.
Theorem 3.5.
Under the conditions of Theorem 3.4, the solution of the schemes (3.1), (2.3), and (2.4) are stable by .
4. Numerical Experiments
Consider the generalized Rosenau equation:
with an initial condition
and boundary conditions
We construct two schemes to (4.1)–(4.3) as nonlinear scheme (2.2) and linearized scheme (3.1). Since we do not know the exact solution of (4.1)–(4.3), we consider the solution on mesh as reference solution and obtain the error estimates on mesh , respectively, for different choices of , where we take . To verify the stability of schemes, we take . The maximal errors are listed on Tables 1, 2, and 3.
Table 1. The errors estimates in the sense of , when and .
Table 2. The errors estimates in the sense of , when and .
Table 3. The errors estimates in the sense of , when and .
We have shown in Theorems 2.2 and 3.1 that the numerical solutions of Scheme (2.2) and Scheme (3.1) satisfy the conservation of energy, respectively. In Figure 1, we give the values of for with fixed for Scheme (2.2). In Figure 2, the values of for Scheme (3.1) are presented. We can see that scheme (2.2) preserves the discrete energy better than scheme (3.1).
From the numerical results, two finite difference schemes of this paper are efficient.
Acknowledgment
This work was supported by the Youth Research Foundation of SWUST (no. 08zx3125).
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