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arXiv 1802.06388math.NAcs.NA
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关于完美匹配层对波动方程的能量稳定不连续伽辽金谱元近似

On energy stable discontinuous Galerkin spectral element approximations of the perfectly matched layer for the wave equation

Ludwig-Maximilian University, Munich(慕尼黑路德维希-马克西米利安大学);Uppsala University(乌普萨拉大学)

Kenneth Duru, Alice-Agnes Gabriel, Gunilla Kreiss

AI总结:

本文提出了一种针对三维和二维线性声波方程的完美匹配层的能量稳定不连续伽辽金谱元方法,通过复杂坐标拉伸推导出高效的无分裂模态PML,证明了连续PML的渐近稳定性,并开发了适用于所有边界条件的DGSEM方法,最后提出了计算PML阻尼系数的方案以确保最优收敛性。

摘要:

我们开发了一种能够证明能量稳定的不连续伽辽金谱元方法(DGSEM)来近似完美匹配层(PML)用于三维和二维线性声波方程,以一阶形式出现,并满足良好定义的线性边界条件。首先,利用广为人知的复坐标拉伸,我们推导出一个高效的无分裂模态PML用于三维声波方程。其次,我们通过推导拉普拉斯空间中的能量估计,证明了在非均匀声学介质中连续PML的渐近稳定性,假设PML阻尼为分段常数。第三,我们开发了使用物理动机的数值通量和惩罚权重的波方程DGSEM,这些权重与所有良好定义的内部和外部边界条件兼容。当PML阻尼消失时,通过构造,我们选择的惩罚参数会生成一个迎风方案和类似于连续能量估计的离散能量估计。第四,为了确保PML阻尼存在时的数值稳定性,有必要系统地将数值通量和元素间及边界过程扩展到PML辅助微分方程中。这在推导离散能量估计时至关重要,类似于连续能量估计。最后,我们提出了一种计算PML阻尼系数的程序,使得PML误差收敛到零,以底层数值方法的最优收敛率。在二维和三维中展示了数值实验,证实了理论结果。

英文摘要:

We develop a provably energy stable discontinuous Galerkin spectral element method (DGSEM) approximation of the perfectly matched layer (PML) for the three and two space dimensional (3D and 2D) linear acoustic wave equations, in first order form, subject to well-posed linear boundary conditions. First, using the well-known complex coordinate stretching, we derive an efficient un-split modal PML for the 3D acoustic wave equation. Second, we prove asymptotic stability of the continuous PML by deriving energy estimates in the Laplace space, for the 3D PML in a heterogeneous acoustic medium, assuming piece-wise constant PML damping. Third, we develop a DGSEM for the wave equation using physically motivated numerical flux, with penalty weights, which are compatible with all well-posed, internal and external, boundary conditions. When the PML damping vanishes, by construction, our choice of penalty parameters yield an upwind scheme and a discrete energy estimate analogous to the continuous energy estimate. Fourth, to ensure numerical stability when PML damping is present, it is necessary to systematically extend the numerical numerical fluxes, and the inter-element and boundary procedures, to the PML auxiliary differential equations. This is critical for deriving discrete energy estimates analogous to the continuous energy estimates. Finally, we propose a procedure to compute PML damping coefficients such that the PML error converges to zero, at the optimal convergence rate of the underlying numerical method. Numerical experiments are presented in 2D and 3D corroborating the theoretical results.

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