On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative
On spatial discretization of evolutionary differential equations on the periodic domain with a mixed derivative
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DOI：
10.1016/j.cam.2019.03.021
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发表时间：
2017-04
期刊：
影响因子：
--
通讯作者：
Shun Sato;Takayasu Matsuo
中科院分区：
文献类型：
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作者：
Shun Sato;Takayasu Matsuo
近年来，各种具有混合导数的演化偏微分方程（PDE）不断涌现，并引起了人们的广泛关注。尽管如此，他们的偏微分方程理论和数值研究仍处于早期阶段。在本文中，我们的目标是在统一的框架内的数值方法，这样的偏微分方程。然而，由于混合导数的存在，我们不能讨论数值方法没有一些适当的重新制定，这是数学上的挑战本身。因此，我们首先提出了一个新的程序重新制定的目标偏微分方程的标准形式的进化方程。这一贡献可能成为一个重要的基础，不仅数值分析，而且偏微分方程理论。为了说明这一点，我们建立了sine-Gordon方程的整体适定性。在此基础上，我们对空间离散化进行了分类和讨论。结果表明，平均差分法适用于混合导数的离散化。
Recently, various evolutionary partial differential equations (PDEs) with a mixed derivative have been emerged and drawn much attention. Nonetheless, their PDE-theoretical and numerical studies are still in their early stage. In this paper, we aim at the unified framework of numerical methods for such PDEs. However, due to the presence of the mixed derivative, we cannot discuss numerical methods without some appropriate reformulation, which is mathematically challenging itself. Therefore, we first propose a novel procedure for the reformulation of target PDEs into a standard form of evolutionary equations. This contribution may become an important basis not only of numerical analysis, but also of PDE-theory. In order to illustrate this point, we establish the global well-posedness of the sine-Gordon equation. After that, we classify and discuss the spatial discretizations based on the proposed reformulation technique. As a result, we show the average-difference method is suitable for the discretization of the mixed derivative.