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

9530937贝利斯将开发、实施和评估基于解的远场展开的波传播问题的非反射边界条件的有效性。将主要考虑瞬变波传播问题，尽管许多技术也将适用于频域。边界条件是首先导出在远场有效的解的收敛或渐近展开式，然后构造消除展开式中的前导阶项的微分算子。通常，这些运算符要求说明假设发出的波的来源。将开发利用内部信息来改善边界条件性能的技术。特别是，将开发自适应边界条件，其中根据内部解的性质动态地和局部地估计原点。还将开发在不增加微分算子的阶数的情况下提高边界条件的阶数的技术。这将通过将非齐次项合并到边界算子中来实现，同样使用内部计算解的性质。最后，将开发适合于阻尼层的远场展开和相关的边界条件，从而允许减小阻尼层的尺寸。将考虑波的数值计算，例如声波和电磁波。这些问题通常由在至少一个方向上没有边界的区域中指定的方程来管理。为了使问题在计算机上表达出来，必须用有界区域来代替无界区域。此外，还需要在有界区域的边缘(边界)施加一些边界条件。这种边界条件可以理想地模拟原始的无界问题。然而，通常情况下，边界处理不会是对无界问题的完美模拟，并且会有从边界发出的虚假反射。这些反射将传播到内部区域，并严重降低计算解的精度。因此，使这些杂散反射最小化对于准确计算是至关重要的。边界条件将根据远场中的波的特性来考虑，即远离产生波的源。目前，对于任何给定的计算，边界条件通常是预先确定的，即在计算开始时，并且不会随着解的时间演变而改变。因此，边界条件没有利用关于内部波的信息。将制定可利用内部信息改进边界条件执行情况的方法。在这种方法中，边界条件会随着时间的推移而变化，通过利用内部信息来适应解的性质。预计，由此产生的边界处理将允许显著减少虚假反射。因此，所提出的新边界条件在计算波传播问题的效率和精度方面提供了显著改进的前景。将考虑在声波、电磁波和弹性波传播方面的应用。

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.