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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
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
By introducing some intermediate variables, the GR-RLW–KdV equation can be written into a multi-symplectic form
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
The scheme (
Next, we are interested in developing schemes possessing the discrete version of LECL (
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.,
For the GR-RLW–KdV equations with periodic boundary conditions, all the proposed structure-preserving algorithms can be written in a form of
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.
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.