Mathematical Sciences: Non-Reflecting Boundary Conditions Based on Far Field Expansions

小行星9530937 波浪无反射边界条件的有效性 传播问题的基础上的远场扩展的 将制定、执行和评估解决办法。 初级瞬态 波的传播问题将被考虑，虽然许多 这些技术也将适用于频域。边界 条件基于第一次推导的收敛或渐进展开 在远场有效的解决方案，然后构建微分 在展开式中消除首阶项的算子。 通常，这些运营商需要指定来源 假定传出波是发散的。 技术将在 该内部信息被用于改善 边界条件 特别是，自适应边界条件将 开发了一种动态和局部估计起源的方法， 解在内部的性质。 增加订单的技术 不增加微分阶数的边界条件 运营商也将发展。 这将通过合并 将非齐次项引入边界算子，再次使用 在内部计算的解决方案。 最后，远场扩展和 适用于阻尼层的相关边界条件为 开发，从而允许减小层的尺寸。 波的数值计算，例如声波和 电磁波将被考虑。 这些问题通常 由在区域中指定的方程控制，这些区域至少在 一个方向。为了使问题在计算机上公式化， 需要将无界区域替换为有界区域。 此外，还必须在边界处施加一些边界条件。 有界区域的边缘（边界）。 该边界条件应 理想地模拟原始无界问题。 然而，一般来说， 边界处理将不会是一个完美的模拟无界 问题，并且将存在从 边界 这些反射将传播到内部区域， 严重降低了计算解的精度。 照经上所 为了精确计算，这些寄生反射必须 最小化 边界条件将根据以下性质进行考虑： 远场中的波，即，远离波浪的源头 生成的. 目前，对于任何给定的计算， 通常预先确定，即，之初 计算，并且不会随着解决方案的及时发展而改变。 因此，在本发明中， 边界条件没有利用关于波的信息， 内政将制定方法，使内部信息能够 用于改善边界条件的性能。 在这 方法，边界条件将随着时间的推移而变化，以适应 通过利用内部信息的性质的解决方案。 是 预计由此产生的边界处理将允许一个显着的 减少寄生反射。 因此，拟议的新边界 条件提供了显著提高效率的前景 波传播问题的计算精度。 应用 声波、电磁波和弹性波的传播 考虑了

9530937 Bayliss The effectiveness of non-reflecting boundary conditions for wave propagation problems which are based on far field expansions of the solution will be developed, implemented and assessed. Primarily transient wave propagation problems will be considered, although many of the techniques will also be applicable in the frequency domain. The boundary conditions are based on first deriving convergent or asymptotic expansions of the solution valid in the far field and then constructing differential operators which annihilate the leading order terms in the expansions. Generally these operators require specification of an origin from which outgoing waves are assumed to emanate. Techniques will be developed in which interior information is used to improve the performance of the boundary condition. In particular, adaptive boundary conditions will be developed in which the origin is estimated dynamically and locally from properties of the solution in the interior. Techniques to increase the order of the boundary condition without increasing the order of the differential operator will also be developed. This will be done by incorporating inhomogeneous terms into the boundary operator, again using properties of the computed solution in the interior. Finally, far field expansions and associated boundary conditions suitable for damping layers will be developed, thereby allowing a reduction in size of the layers. The numerical computation of waves, for example, acoustic and electromagnetic waves, will be considered. These problems are typically governed by equations specified in regions which are unbounded in at least one direction. In order for the problem to be formulated on a computer it is necessary that the unbounded region be replaced by a bounded region. Furthermore, it is necessary that some boundary condition be imposed at the edge (boundary) of the bounded region. This boundary condition should ideally simulate the ori ginal unbounded problem. However, in general the boundary treatment will not be a perfect simulation of the unbounded problem and there will be spurious reflections emanating from the boundary. These reflections will propagate into the interior region and seriously degrade the accuracy of the computed solution. Thus, it is essential for accurate computation that these spurious reflections be minimized. Boundary conditions will be considered based on properties of the waves in the far field, i.e., far from the sources where the waves are generated. Currently for any given computation the boundary conditions are generally determined in advance, i.e., at the beginning of the computation, and are not changed as the solution evolves in time. Thus, the boundary conditions make no use of information about the waves in the interior. Methodologies will be developed in which interior information can be used to improve the performance of the boundary conditions. In this approach, the boundary conditions will change as time evolves, adapting to the nature of the solution by making use of interior information. It is anticipated that the resulting boundary treatment will allow a significant reduction in spurious reflections. Thus, the proposed new boundary conditions offer the prospect of significant improvement in efficiency and accuracy for the computation of wave propagation problems. Applications to acoustic, electromagnetic and elastic wave propagation will be considered.