几何数值积分器用于Hunter-Saxton型方程
Geometric numerical integrators for Hunter-Saxton-like equations
AI总结：
本文提出新的几何数值积分器用于Hunter-Saxton型方程，通过新的多辛 formulations 和已知的哈密顿结构。通过显式欧拉框方案和离散变分导数方法展示多辛离散化和哈密顿保持离散化，并统一处理边界条件，验证了所提数值积分器的有效性。
摘要：
我们通过新的多辛 formulations 和已知问题的哈密顿结构，提出了 Hunter--Saxton 型方程的新几何数值积分器。我们考虑了 Hunter--Saxton 方程、修正的 Hunter--Saxton 方程以及两组分 Hunter--Saxton 方程。基于这些新 formulations 的多辛离散化通过显式欧拉框方案进行示例，而哈密顿保持离散化则通过离散变分导数方法进行示例。我们通过统一的方式解释并证明了边界条件的正确处理，这对于这些方程的正确数值实现是必要的，此前文献中从未明确阐明。最后，数值实验展示了所提数值积分器的良好行为。
英文摘要：
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.
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