The computation of water waves modelled by Nekrasov's equation
The computation of water waves modelled by Nekrasov's equation
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用 Nekrasov 方程模拟的水波计算
DOI：
10.1137/0730054
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发表时间：
1993
影响因子：
2.9
通讯作者：
I. Graham
中科院分区：
文献类型：
--
作者：
G. Chandler;I. Graham
Nekrasov的积分方程，描述水波的几乎极端的形式，数值求解。该方法包括应用一个简单的求积规则的原始方程的重新排列的版本。强梯度网格被用来解决一个预期的边界层的解决方案。对于基于梯形法则的方法，利用全局分歧理论证明了，对于固定的离散参数n，正数值解存在一个连续的分支。这些都是参数化的$\mu $一个自然的参数发生在原来的积分方程。对于固定的$\mu $，集体紧性参数然后证明这些解决方案的连续收敛，因为网格是细化的（即，如$n \to \infty $）。使用高阶求积规则的数值实验报告。这些表明，该方法能够检测宽度$O（{{70} \mathord{\left/ {\vphantom {..的区域中最大高度约为$0.37^ \circ $的边界层和吉布斯现象类型振荡。
Nekrasov’s integral equation, describing water waves of almost extreme form, is solved numerically. The method consists of applying a simple quadrature rule to a rearranged version of the original equation. Strongly graded meshes are used to resolve an expected boundary layer in the solution. For methods based on the trapezoidal rule, global bifurcation theory is used to prove, for fixed discretization parameter n, the existence a continuous branch of positive numerical solutions. These are parametrized by $\mu $ a natural parameter occurring in the original integral equation. For fixed $\mu $, collective compactness arguments then prove subsequential convergence of these solutions as the mesh is refined (i.e., as $n \to \infty $). Numerical experiments using higher-order quadrature rules are reported. These reveal that the method is capable of detecting a boundary layer and Gibbs phenomenon type oscillations of maximum height about $0.37^ \circ $ in a region of width $O({{70} \mathord{\left/ {\vphantom {...