摘要
本文考虑了对称正则长波(SRLW)方程的多辛算法。辛算法是从辛几何观点出发，利用变分原理构造的具有保持原Hamilton系统辛几何结构性质的一种算法。其基本思想：首先，利用正则变换，构造偏微分方程的多辛方程组。然后，利用多辛算法离散此多辛方程组，得到它的多辛格式，要求所得到的多辛格式满足离散形式的多辛守恒律。这是最为关键的一点。最后，通过数值实验检验算法的有效性与优越性。
我们通过对SRLW方程作正则变换，得到了它的一个正则方程组及其几个守恒律。用多辛方法离散此方程组，得到了它的一些多辛格式，证明了它们具有离散形式的多辛守恒律，并且它的局部能量与动量守恒律的离散形式具有较高阶的误差精度。对中点格式，通过消去中间变量，得到不含中间变量的与原格式等价的多辛Preissman格式。通过对多辛中点格式分别作适当的不同的变化，我们分别得到了满足离散局部能量守恒律格式与离散局部动量守恒律格式，但这两格式除线性情形外均不是多辛格式，对这两格式我们也得到了它的等价格式。我们用大量数值实验验证了所构造的格式的有效性与长时间的数值稳定性，它们还能很好地模拟原孤立波的波形与具有较高的能量精度。
最后，我们还简要介绍了多辛Fourier拟谱方法，然后把它应用到SRLW方程的多辛方程组得到了它的多辛Fourier拟谱格式。我们用数值实验验证了它的有效性。数值结果表明多辛Fourier拟谱格式较多辛中点格式具有较高的精度。
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.
引文
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