论文标题
有损媒体的波方程的数值分析遵守频率功率定律
Numerical analysis of a wave equation for lossy media obeying a frequency power law
论文作者
论文摘要
我们研究具有非局部时间分数阻尼项的波方程，该术语模拟了以频率依赖力定律为特征的声学衰减的影响。首先，我们证明存在该方程式的独特解决方案，并特别注意分数衍生物的处理。然后，我们基于空间中的有限元方法以及卷积正交和二阶阶段的中心时间差异来得出一个明确的时间步进方案。我们对混合时间离散化的完整误差分析，进而对完全时空离散的方案进行。对平滑解决方案和奇异性的解决方案均给出了错误估计，该解决方案是$ t = 0 $的类型的奇异性，对于涉及分数时间衍生物的方程式是典型的。提出了许多数值结果以支持误差分析。
We study a wave equation with a nonlocal time fractional damping term that models the effects of acoustic attenuation characterized by a frequency dependence power law. First we prove existence of a unique solution to this equation with particular attention paid to the handling of the fractional derivative. Then we derive an explicit time stepping scheme based on the finite element method in space and a combination of convolution quadrature and second order central differences in time. We conduct a full error analysis of the mixed time discretization and in turn the fully space time discretized scheme. Error estimates are given for both smooth solutions and solutions with a singularity at $t = 0$ of a type that is typical for equations involving fractional time-derivatives. A number of numerical results are presented to support the error analysis.
