多辛Preissman格式及其等价格式的数值实现
Numerical Implementation of the Multisymplectic Preissman Scheme and Its Equivalent Schemes
摘要：
我们分析了KdV方程周期边界条件下的多辛Preissman格式，并指出Preissman格式的非线性代数系统的迭代求解的不收敛性是由于引入了势函数。在周期性边界条件的基础上，增加了一个人工数值条件。增加的边界条件使多辛Preissman格式的数值实现变得实用，并被证明不会改变KdV方程的数值解。在此基础上，我们导出了一些新的格式，这些格式不受人工边界条件的限制，而且由于计算量小，所需计算机内存少，比Preissman格式更有效。通过去掉辅助变量，我们还得到了KdV方程的两个格式，一个是12点格式，另一个是8点格式。作为副产品，我们提出了两个新的显式格式，它们不是多辛的，但仍然具有显着的数值稳定性。
英文摘要：
We analyze the multisymplectic Preissman scheme for the KdV equation with the periodic boundary condition and show that the unconvergence of the widely-used iterative methods to solve the resulting nonlinear algebra system of the Preissman scheme is due to the introduced potential function. A artificial numerical condition is added to the periodic boundary condition. The added boundary condition makes the numerical implementation of the multisymplectic Preissman scheme practical and is proved not to change the numerical solutions of the KdV equation. Based on our analysis, we derive some new schemes which are not restricted by the artificial boundary condition and more efficient than the Preissman scheme because of less computing cost and less computer storages. By eliminating the auxiliary variables, we also derive two schemes for the KdV equation, one is a 12-point scheme and the other is an 8-point scheme. As the byproducts, we present two new explicit schemes which are not multisymplectic but still have remarkable numerical stable property.
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