Temporally Uniform Grid Convergence of Discrete Approximations and Numerical Simulations in the Problems of Wave Propagation over Unbounded Domains

在数值模拟波浪在无界区域上的传播过程中，两个公认的主要困难是在长时间间隔内误差的积累和截断区域并随后在外部人工边界处设置人工边界条件（ABC）的必要性。这两个问题原来是密切相关的。在PI与合作者在标量波动方程框架内进行的先前工作中，我们使用了固有的三维空隙现象，并开发了一种修改任何适当离散方案的方法，以便完全消除长期误差积累，同时保留方案的所有原始属性（例如，准确度）。此外，该程序允许一个有限的计算域取代原来的无限域，这导致获得高精度的非局部非定常ABC。这些ABC是直接为问题的离散形式而建立的，它们的时间非局部性是固定的和有限的，并且它们具有完全的几何普适性。拟议项目的主要目标是将上述方法推广到具有实际意义的波型模型，特别是麦克斯韦方程（电磁波）和线性化欧拉方程和全势方程（声波）。这一目标的可达性是由这些方程的解具有波的尖后锋这一事实来说明的（腔隙的表现），这与波动方程的解所显示的行为完全相同。无界域上的波的数值模拟有许多应用，从飞机和地面车辆的电磁波散射从雷达技术到天线设计，再到计算机身和飞机喷气发动机产生的声场，以降低机场周围和乘客舱内的噪音水平。我们期望得到的结果将有利于上述应用领域。事实上，我们预计创建一个通用的框架，将允许一个修改广泛的类已经存在的和证明的方法，使两个额外的关键能力-长期集成和精确计算的无限域波场截断域。此外，随着拟议研究的展开，它必然包括与物理学家和工程师的沟通和合作，以及准备课程材料和培训年轻研究人员。

Two major well-recognized difficulties in numerical simulation of wavespropagating over unbounded domains are the accumulation of error during long time intervals and necessity to truncate the domain and subsequently set the artificial boundary conditions (ABCs) at the external artificialboundary. These two issues turn out to be closely related. In the previouswork of the PI with collaborators that has been done in the framework ofthe scalar wave equation, we have used the inherently three-dimensional phenomenon of lacunae and developed a methodology that modifies any appropriate discrete scheme so that the long-term error buildup is fully eliminated while all of the original properties of the scheme (e.g., order of accuracy) are preserved. Moreover, the procedure allows one to replace the original infinite domain by a finite computational domain, which leads to obtaining highly accurate non-local unsteady ABCs. These ABCs are built directly for the discrete formulation of the problem, their temporal non-locality is fixed and limited, and they possess full geometric universality. The key objective of the proposed project is to extend the aforementioned methodology to wave-type models of practical interest, in particular, the Maxwell's equations (electromagnetic waves) and the linearized Euler's and full-potential equations (acoustic waves). The attainability of this goal is accounted for by the fact that the solutions to these equations have sharp aft fronts of the waves (manifestation of lacunae), which is the exact same behavior as displayed by the solutions to the wave equation.Numerical simulation of waves on unbounded domains has numerous applications that range from scattering of electromagnetic waves from aircraft and ground vehicles (radar technology) to antenna design to calculation of the acousticfields produced by the airframe and airplane's jet engines for the purpose of reducing the noise levels around airports, as well as inside the passenger compartments. The results that we expect to obtain are going to benefit the foregoing applied areas. Indeed, we anticipate the creation of a universal framework that would allow one to modify a wide class of already existing and proven methods so that to enable two additional crucial capabilities -- long-term integration and accurate computation of infinite-domain wave fields on truncated domains. Besides, as the proposed research unfolds, it will necessarily include communication and collaboration with physicists and engineers, as well as preparation of course materials and training young researchers.