一维波动方程间断Galerkin半离散化的局部解与滤波机制
Localized solutions and filtering mechanisms for the discontinuous Galerkin semi-discretizations of the 1-d wave equation
AI总结：
针对一维波动方程的P1间断Galerkin半离散化，通过傅里叶分析揭示物理分量与虚假分量的相互作用，构建任意小速度的波包，并发展滤波机制恢复均匀传播速度，应用于控制问题的数值逼近。
摘要：
我们对通过均匀网格上连续波动方程的P1间断Galerkin（DG）逼近得到的一维半离散波动方程进行了完整的傅里叶分析。所得系统表现出两种类型分量的相互作用：物理分量和虚假分量，后者与数值解允许的可能间断性有关。每个色散关系包含临界点，其中相应的群速度为零。基于先前的构造，我们严格构建了具有任意小传播速度的波包，这些波包集中在物理分量或虚假分量上。我们还开发了旨在恢复连续解均匀传播速度的滤波机制。最后，还介绍了在控制问题的数值逼近问题中的一些应用。
英文摘要：
We perform a complete Fourier analysis of the semi-discrete 1-d wave equation obtained through a P1 discontinuous Galerkin (DG) approximation of the continuous wave equation on an uniform grid. The resulting system exhibits the interaction of two types of components: a physical one and a spurious one, related to the possible discontinuities that the numerical solution allows. Each dispersion relation contains critical points where the corresponding group velocity vanishes. Following previous constructions, we rigorously build wave packets with arbitrarily small velocity of propagation concentrated either on the physical or on the spurious component. We also develop filtering mechanisms aimed at recovering the uniform velocity of propagation of the continuous solutions. Finally, some applications to numerical approximation issues of control problems are also presented.
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