When modeling wave propagation, truncation of the computational domain to a manageable size requires nonreflecting boundaries. To construct such a boundary condition on one side of a rectangular domain for a finite-difference discretization of the acoustic wave equation in the frequency domain, the domain is extended on that one side to infinity. Constant extrapolation in the direction perpendicular to the boundary provides the material properties in the exterior. Domain decomposition can split the enlarged domain into the original one and its exterior. Because the boundary-value problem for the latter is translation-invariant, the boundary Green functions obey a quadratic matrix equation. Selection of the solvent that corresponds to the outgoing waves provides the input for the remaining problem in the interior. The result is a numerically exact nonreflecting boundary condition on one side of the domain. When two nonreflecting sides have a common corner, the translation invariance is lost. Treating each side independently in combination with a classic absorbing condition in the other direction restores the translation invariance and enables application of the method at the expense of numerical exactness. Solving the quadratic matrix equation with Newton’s method turns out to be more costly than solving the Helmholtz equation and may select unwanted incoming waves. A proposed direct method has a much lower cost and selects the correct branch. A test on a 2D acoustic marine seismic problem with a free surface at the top, a classic second-order Higdon condition at the bottom, and numerically exact boundaries at the two lateral sides demonstrates the capability of the method. Numerically exact boundaries on each side, each computed independently with a free-surface or Higdon condition, provide even better results.

中文翻译：

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应用于声学亥姆霍兹方程的数值精确的非反射边界条件

在模拟波传播时，将计算域截断到可管理的大小需要非反射边界。为了在频域中对声波方程进行有限差分离散，在矩形域的一侧构造这样的边界条件，该域在该侧扩展到无穷大。在垂直于边界的方向上的恒定外推提供了外部的材料特性。域分解可以将扩大后的域拆分为原始域和外部域。因为后者的边值问题是平移不变的，所以边界格林函数服从二次矩阵方程。对应于出射波的溶剂的选择提供了内部剩余问题的输入。结果是域一侧的数值精确的非反射边界条件。当两个非反射边有一个公共角时，平移不变性丢失。独立处理每一侧并结合另一个方向上的经典吸收条件恢复平移不变性，并使该方法的应用能够以数值准确性为代价。事实证明，使用牛顿法求解二次矩阵方程比求解亥姆霍兹方程成本更高，并且可能会选择不需要的入射波。建议的直接方法具有低得多的成本并选择正确的分支。对顶部为自由表面、底部为经典二阶 Higdon 条件的二维声学海洋地震问题的测试，两侧的数值精确边界证明了该方法的能力。每边的数字精确边界，每个边界都使用自由表面或 Higdon 条件独立计算，提供更好的结果。