High-Order Embedded Interface Methods for Wave-Problems

我们提出了一种新的稳定的高阶有限差分方法，适用于解决涉及许多接口和显着的几何复杂性的波动问题，这些并发症是通过嵌入到笛卡尔网格的计算问题和制定的方法，这样的位置的材料接口以及物理性质的解决方案的整个接口是accountforproperly顺序的计划。交错网格以及非交错网格方法将探讨与重点发展一个严格的数学基础，这些schemes，以确保鲁棒性和统一的稳定性，为所有网格大小和嵌入的几何形状。嵌入方法的吸引人的特性，如模拟移动界面的能力和引入虚拟界面以增强并行性能，将被用来模拟电磁学、声学、地震学和弹性学中的各种波动问题。人们对精确有效地解决波动主导问题（如声学和电磁学问题）的兴趣日益增加，在很长一段时间内，产生了对制定高阶精确计算方法的兴趣。在这项工作中，我们建议开发一类新的计算技术，特别是针对可靠和强大的建模问题的现实规模和复杂性，例如，电磁和声学噪声传播及其环境影响的建模，以及石油工业感兴趣的地下波的建模。所提出的方法在保持非常简单的计算结构而不牺牲精度方面是独一无二的，因此，克服了与现有方法相关的许多众所周知的困难，现有方法需要某种自动生成计算网格，计算。这些建议的发展将使非常复杂和现实的情况下建模，并将与高性能的计算设施相结合的方法是非常适合的，允许建模和分析复杂的波为主的问题，在各种领域感兴趣的工程师和科学家。

We propose to develop a new family of stable high-order finitedifference methods suitable for the solution of wave problemsinvolving many interfaces and significant geometric complexity.These complications are addressed by embedding the computationalproblem into a Cartesian grid and formulating methods such thatthe position of the material interfaces as well as the physicalproperties of the solution across the interface is accounted forproperly to the order of the scheme. Staggered grid as well asnon-staggered grid methods will be explored with the emphasis onthe development of a rigorous mathematical foundation for theseschemes to ensure robustness and uniform stability for all grid sizes and embedded geometries. Appealing properties of embeddingmethods such as the ability to model moving interfaces and theintroduction of virtual interfaces to enhance parallel performancewill be exploited to model a variety of wave problems inelectromagnetics, acoustics, seismology, and elasticity.The increasing interest in the accurate and efficient solutionof wave-dominated problems, e.g. problems in acoustic and electromagnetics, over very long periods of time has spawned an interest in the formulation of high-order accurate computationalmethods for such problems. In this effort we propose to developa new class of computational techniques, specifically aimed atthe reliable and robust modeling of problems of a realisticsize and complexity, e.g., the modeling of the propagation ofelectromagnetic and acoustic noise and its environmental impact, andthe modeling of underground waves of interest to the oil industry.The proposed methods are unique in maintaining a very simple computational structure without sacrificing the accuracy, henceovercoming a number of well known difficulties associated withexisting methods which require some kind of automated generation ofthe computational grid on which the solution is computed.These proposed developments will enable the modeling ofvery complex and realistic scenarios and will, in combination with high-performance computing facilities for which themethods are well suited, allow for the modeling and analysisof complex wave dominated problems in a variety of areas of interestto engineers and scientists.