Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems
Combining finite element space-discretizations with symplectic time-marching schemes for linear Hamiltonian systems
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DOI：
10.3389/fams.2023.1165371
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发表时间：
2023-04
期刊：
影响因子：
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通讯作者：
Bernardo Cockburn;Shukai Du;M. Sánchez
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文献类型：
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作者：
Bernardo Cockburn;Shukai Du;M. Sánchez
我们提供了一个简短的介绍设计的一种特殊类型的方法，数值逼近的解决方案的哈密顿偏微分方程。这些方法使用Galerkin空间离散化，这导致在一个系统的常微分方程显示离散版本的哈密顿结构的原始系统。所得到的系统的常微分方程，然后离散的辛时间推进方法。这种组合的结果在高阶精确，完全离散的方法，可以保持定义的ODE系统的哈密顿量的不变量。我们限制我们的注意力线性哈密顿系统，因为主要结果可以很容易地得到，直接，并适用于许多哈密顿系统的实际利益，包括声学，弹性动力学，电磁学。在简要描述了我们感兴趣的哈密顿系统之后，我们简要介绍了线性常微分方程组的辛时间推进方法，该方法不需要任何背景知识。然后，我们描述的情况下，有限差分空间离散的使用，并集中在流行的Yee计划（1966年）的电磁。最后，我们考虑有限元空间离散化的情况。重点放在全离散格式的守恒性质。最后，我们描述了正在进行的工作。
We provide a short introduction to the devising of a special type of methods for numerically approximating the solution of Hamiltonian partial differential equations. These methods use Galerkin space-discretizations which result in a system of ODEs displaying a discrete version of the Hamiltonian structure of the original system. The resulting system of ODEs is then discretized by a symplectic time-marching method. This combination results in high-order accurate, fully discrete methods which can preserve the invariants of the Hamiltonian defining the ODE system. We restrict our attention to linear Hamiltonian systems, as the main results can be obtained easily and directly, and are applicable to many Hamiltonian systems of practical interest including acoustics, elastodynamics, and electromagnetism. After a brief description of the Hamiltonian systems of our interest, we provide a brief introduction to symplectic time-marching methods for linear systems of ODEs which does not require any background on the subject. We describe then the case in which finite-difference space-discretizations are used and focus on the popular Yee scheme (1966) for electromagnetism. Finally, we consider the case of finite-element space discretizations. The emphasis is placed on the conservation properties of the fully discrete schemes. We end by describing ongoing work.