论文标题
通过波方程来扩展和分析Helmholtz方程的迭代溶液
Extensions and Analysis of an Iterative Solution of the Helmholtz Equation via the Wave Equation
论文作者
论文摘要
在本文中，我们扩展了Waveholtz迭代的分析 - 用于helmholtz方程溶液的时域迭代方法。我们扩展了对能源保护问题的先前分析，并证明了在单个空间维度下具有阻抗边界条件的问题的波霍尔兹迭代的收敛性。然后，我们考虑在任何空间尺寸中阻尼的内部Dirichlet/Neumann问题，并表明，对于足够的水平阻尼Waveholtz迭代在许多迭代中都会收敛于许多与频率无关的迭代。最后，我们为一个高阶时间步变方案的家族进行了对波霍尔兹迭代的离散分析。我们表明，离散波霍兹迭代的固定点通过选择的时间stepper的顺序将离散的Helmholtz解决方案收敛到离散的Helmholtz解决方案。我们提出数值示例，并证明可以通过仔细分析离散迭代以及更新的正交公式从Waveholtz解决方案中完全消除时间离散化误差。
In this paper we extend analysis of the WaveHoltz iteration -- a time-domain iterative method for the solution of the Helmholtz equation. We expand the previous analysis of energy conserving problems and prove convergence of the WaveHoltz iteration for problems with impedance boundary conditions in a single spatial dimension. We then consider interior Dirichlet/Neumann problems with damping in any spatial dimension, and show that for a sufficient level of damping the WaveHoltz iteration converges in a number of iteration independent of the frequency. Finally, we present a discrete analysis of the WaveHoltz iteration for a family of higher order time-stepping schemes. We show that the fixed-point of the discrete WaveHoltz iteration converges to the discrete Helmholtz solution with the order of the time-stepper chosen. We present numerical examples and demonstrate that it is possible to completely remove time discretization error from the WaveHoltz solution through careful analysis of the discrete iteration together with updated quadrature formulas.
