Geometric numerical integrators for Hunter-Saxton-like equations
Geometric numerical integrators for Hunter-Saxton-like equations
复制标题
Hunter-Saxton 类方程的几何数值积分器
DOI：
10.1007/s13160-017-0252-1
复制
发表时间：
2017
影响因子：
0.9
通讯作者：
Matsuo Takayasu
中科院分区：
文献类型：
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作者：
Miyatake Yuto;Cohen David;Furihata Daisuke;Matsuo Takayasu
我们提出了新的几何数值积分的Hunter-Saxton类方程通过新的多辛公式和已知的哈密顿结构的问题。我们考虑了Hunter-Saxton方程、修正的Hunter-Saxton方程和双分量Hunter-Saxton方程。基于这些新的配方的问题的多辛离散的例子通过显式欧拉盒计划，和Hamilton保持离散的例子通过离散变分导数方法。我们以统一的方式解释和证明边界条件的正确处理。这是必要的适当的数值实现这些方程，从来没有明确澄清的文献中，尽我们所知。最后，数值实验表明所提出的数值积分器的良好行为。
We present novel geometric numerical integrators for Hunter–Saxton-like equations by means of new multi-symplectic formulations and known Hamiltonian structures of the problems. We consider the Hunter–Saxton equation, the modified Hunter–Saxton equation, and the two-component Hunter–Saxton equation. Multi-symplectic discretisations based on these new formulations of the problems are exemplified by means of the explicit Euler box scheme, and Hamiltonian-preserving discretisations are exemplified by means of the discrete variational derivative method. We explain and justify the correct treatment of boundary conditions in a unified manner. This is necessary for a proper numerical implementation of these equations and was never explicitly clarified in the literature before, to the best of our knowledge. Finally, numerical experiments demonstrate the favourable behaviour of the proposed numerical integrators.