Cai Jia-Xiang, Hong Qi, Yang Bin. Local structure-preserving methods for the generalized Rosenau–RLW–KdV equation with power law nonlinearity. Chinese Physics B, 2017, 26(10): 100202
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Local structure-preserving methods for the generalized Rosenau–RLW–KdV equation with power law nonlinearity
Cai Jia-Xiang1, †, Hong Qi2, Yang Bin1
School of Mathematical Science, Huaiyin Normal University, Huaian 223300, China
Graduate School of China Academy of Engineering Physics, Beijing 100083, China
Local structure-preserving algorithms including multi-symplectic, local energy- and momentum-preserving schemes are proposed for the generalized Rosenau–RLW–KdV equation based on the multi-symplectic Hamiltonian formula of the equation. Each of the present algorithms holds a discrete conservation law in any time–space region. For the original problem subjected to appropriate boundary conditions, these algorithms will be globally conservative. Discrete fast Fourier transform makes a significant improvement to the computational efficiency of schemes. Numerical results show that the proposed algorithms have satisfactory performance in providing an accurate solution and preserving the discrete invariants.
A nonlinear wave phenomenon is one of the important areas of scientific research. There are various mathematical models describing the wave behaviors. The well-known KdV equation was put forward as a model for small-amplitude long waves on the surface of water in a channel and the RLW equation was originally introduced to describe the behavior of the undular bore. Both KdV and RLW equations have been well studied theoretically and numerically.[1,2] Since the case of wave–wave and wave–wall interactions cannot be described by the KdV equation, the Rosenau equation was proposed, which describes the dynamic of a dense discrete system.[3] For the further consideration of nonlinear waves, the viscous terms uxxx and uxxt need to be included in the Rosenau equation respectively. The resulting equations are usually called Rosenau–KdV and –RLW equations, respectively. The initial boundary value problems have been studied numerically in the past few years.[4–9] In this paper, we are interested in developing numerical methods for the following generalized Rosenau–RLW–KdV (GR-RLW–KdV) equation with a power law nonlinearity
subjected to an initial condition , and periodic boundary conditions, where p is a positive integer. The parameter b = 0 gives the generalized Rosenau–RLW (GR-RLW) equation and parameter σ = 0 reads the generalized Rosenau–KdV (GR-KdV) equation. Since the GR-RLW–KdV equation includes a power law nonlinearity, to simulate it well, the numerical method should have some excellent properties such as good accuracy in solution and long-term stability. To the best of our knowledge, there are few numerical methods for the equation in the literature.
Generally, the performance of a numerical method is affected by not only the accuracy of the method but also other factors such as the conservative approximation property of the method. Better solutions can be expected from numerical schemes which have effective conservative approximation properties rather than the ones which have nonconservative properties.[10–13] Some conservative methods have been proposed for the Rosenau-type equations.[5–8] The preserving discrete global conservation laws of these methods depend on the boundary conditions. That is, if the original problem does not have appropriate boundary conditions, these conservative methods cannot be applied to it. Actually, the GR-RLW–KdV equation admits local conservation laws, such as the multi-symplectic conservation law, local energy conservation law, and momentum conservation law. These conservation laws are independent of the boundary conditions and can produce richer information of the original system. Naturally, it is desired to develop numerical methods preserving the discrete version of the local conservation laws. Note that as the boundary conditions are appropriate, these methods will be globally conservative. The main purpose of this paper is to develop some local structure-preserving methods for the GR-RLW–KdV equation based on its multi-symplectic Hamiltonian formula, and carry out some numerical experiments.
To show the derivation of the schemes conveniently, we describe the computational grid and give some definitions of finite difference operators. Consider a mesh as a uniform partition of the solution domain by the knots xj with , . Let unj be a numerical approximation to where τ is an increment in time. Define shift operators , and finite difference operators
For simplicity, in most of the equations we present, notation unj is replaced by u, and one of the indices is held constants, in which case, we drop it from the notation.
2. Structure-preserving algorithms
By introducing some intermediate variables, the GR-RLW–KdV equation can be written into a multi-symplectic form
where
is a scalar-valued smooth function, , and the two skew-symmetric matrices , with non-zero entries are given by , , , , , , and . The system (2) possesses three local conservation laws,[14] namely multi-symplectic conservation law (MSCL)
local energy conservation law (LECL)
and local momentum conservation law (LMCL)
where splitting matrices and satisfy
It is obvious that the three conservation laws are local, i.e., they are independent of the boundary conditions. With appropriate boundary conditions such as periodic/homogeneous boundary conditions, integrating them over the spatial region gives global conservation laws. Since the local conservation laws are more essential than the global ones, we propose some methods preserving the discrete version of MSCL, LECL, and LMCL, respectively.
The concepts of the multi-symplectic structure and the multi-symplectic integrator were introduced in Ref. [14], which have wide applications.[15–19] Multi-symplectic (MS) box/Preissman integrator
is very popular for the multi-symplectic Hamiltonian partial differential equation (PDE). Applying it to the multi-symplectic Hamiltonian formulation of the GR-RLW–KdV equation, and then eliminating the auxiliary variables gives a scheme
The scheme (8) possesses a discrete MSCL where and , which is a discrete version of the MSCL (3).
Next, we are interested in developing schemes possessing the discrete version of LECL (4) and LMCL (5) for the GR-RLW–KdV equation. We discretize the system (2) as follows:
where represents a discrete gradient method in time. By some simple derivations, we prove that the scheme (9) preserves a discrete LECL
Also, we make a discretization
for the system (2) where is a discrete gradient method in space. The scheme has a discrete LMCL
In this paper, the discrete gradient method is taken as the averaged vector field.[20] Applications of LECL (9) and LMCL (11) integrators to the multi-symplectic formulations of the GR-RLW–KdV equation give respectively the following local energy-preserving (LEP) scheme
and local momentum-preserving (LMP) scheme
where
3. Conservative properties and fast computation
Assume that the GR-RLW–KdV equation is imposed on periodic or homogeneous boundary conditions. Then the LEP and LMP schemes conserve discrete global energy and momentum conservation laws respectively, i.e., and , where
By summing the schemes (8), (13), and (14) over all index js, we prove that the schemes have the same discrete mass conservative law where .
For the GR-RLW–KdV equations with periodic boundary conditions, all the proposed structure-preserving algorithms can be written in a form of
where both A and B are constant circulant matrices. The nonlinear system (15) can be solved by a fix-point iteration method with initial iterative vector , i.e.,
Note that the constant circulant matrix A can be represented as
where is the matrix of discrete Fourier transform coefficients with entries given by , , and the diagonal matrix with . The property (17) is very helpful to improve the computational efficiency of equation (16), i.e., it can be solved explicitly.
4. Numerical results
In the following, we carry out some numerical experiments to show the performance of the proposed schemes in term of the accuracy of solution and preservation of the discrete invariants.
5. Concluding remarks
The GR-RLW–KdV equation can be reformed into a multi-symplectic Hamiltonian formula, which admits MSCL, LECL, and LMCL. These conservation laws are local, i.e., they are independent of boundary conditions. For the problem with appropriate boundary conditions, one can obtain global conservation laws. Therefore, the local conservation laws can produce more information of the original problem than the global ones. In this work, we proposed some local structure-preserving algorithms preserving the discrete version of MSCL, LECL, and LMCL respectively for the GR-RLW–KdV equation. With periodic/homogeneous boundary conditions, the present schemes are globally conservative. By the discrete fast Fourier transform, the computational efficiency of the proposed schemes is improved. Some numerical experiments are carried out to test the numerical performance of the present schemes.
