论文标题
Helmholtz方程的时间域预处理
A time-domain preconditioner for the Helmholtz equation
论文作者
论文摘要
波动方程的时谐解解决方案可以在频率或时域中计算。在频域中，一个人求解了离散的Helmholtz方程，而在时域则寻求离散波动方程的周期性解决方案，例如通过使用时间谐波的强迫术语模拟很长时间。时间域方法的缺点是该解决方案受时间离散错误的影响，并且无法自由选择空间离散化，因为它是从时间域方案继承的。在这项工作中，我们解决了这些问题。给定满足某些特性的无限线性系统，构建了矩阵复发关系，以便在极限中获得确切的离散解决方案。通过迭代大量有限的次数，获得了近似解决方案，与Helmholtz方程的时间域方法类似。为了改善收敛性，该过程用作GMRE的预处理，并且时谐波强迫项乘以平滑的窗口函数。该结构应用于Helmholtz方程的紧凑型模板有限差异化离散化，以前没有时间域求解器可用。所得求解器的优点是相对简单，记忆要求较小和合理的计算时间。
Time-harmonic solutions to the wave equation can be computed in the frequency or in the time domain. In the frequency domain, one solves a discretized Helmholtz equation, while in the time domain, the periodic solutions to a discretized wave equation are sought, e.g. by simulating for a long time with a time-harmonic forcing term. Disadvantages of the time-domain method are that the solutions are affected by temporal discretization errors and that the spatial discretization cannot be freely chosen, since it is inherited from the time-domain scheme. In this work we address these issues. Given an indefinite linear system satisfying certain properties, a matrix recurrence relation is constructed, such that in the limit the exact discrete solution is obtained. By iterating a large, finite number of times, an approximate solution is obtained, similarly as in a time-domain method for the Helmholtz equation. To improve the convergence, the process is used as a preconditioner for GMRES, and the time-harmonic forcing term is multiplied by a smooth window function. The construction is applied to a compact-stencil finite-difference discretization of the Helmholtz equation, for which previously no time-domain solver was available. Advantages of the resulting solver are the relative simplicity, small memory requirement and reasonable computation times.
