Symplectic Algorithm For The Symmetric Regularized Long Wave Equation

We consider some multi-symplectic schemes for SRLW equation in the paper. Symplectic algorithm sets forth symplectic geometry and makes use of variation principal. It requests that schemes should maintain symplectic geometric traits of the original Hamiltonian system. Its main idea is as follows: Firstly, we construct multi-symplectic equations for PDEs through canonical transformations. Secondly, we discrete multi-symplectic equations with multi-symplectic schemes which must preserve discretic multi-symplectic conservation law. This is one of the most key points. Lastly, we should perform numerical experiments to verify the validities and superiorities of the schemes we have established.Canconical equations for SRLW equation are presented which possess some conservation laws by canonical transformations in this paper. We present some multi-symplectic schemes for the equations. They preserve discretic multi-symplectic conservation law exactly. Errors orders of their local discretic energy and local discretic momentum conservation laws are very high. We eliminate middle variables and amount to a single variable multi-symplectic scheme for mid-point scheme. We call it Preissman scheme. We come up with preserving local discretic energy conservation law scheme if we act some proper transformations on mid-point scheme; Preserving local discrete momentum conservation law scheme is also presented here in a similar way. However, neither is these two schemes multi-symplectic scheme except linear case. We perform a lot of experiments. Experiments show that schemes we construct are available. They also verify that multi-symplectic scheme are capable of preserving original solitary wave shape. They also prove that multi-symplectic schemes have long time numerical behavior as well as high energy accurcy.We introduce multi-symplectic Fourier pseudo-spectral method simply in the end. Then we apply it to multi-symplectic equations of SRLW and get a multi-symplectic Fourier pseudo-spectral scheme for it. We also carry out some experiments to illustrate its validity. Numerical results show that multi-symplectic Fourier pseudo-spectral scheme has higher accuracy than multi-symplectic Preissman scheme, either.