High-Order Embedded Interface Methods for Wave-Problems

Brown University, Providence RI

Investigators

Abstract

We propose to develop a new family of stable high-order finite difference methods suitable for the solution of wave problems involving many interfaces and significant geometric complexity. These complications are addressed by embedding the computational problem into a Cartesian grid and formulating methods such that the position of the material interfaces as well as the physical properties of the solution across the interface is accounted for properly to the order of the scheme. Staggered grid as well as non-staggered grid methods will be explored with the emphasis on the development of a rigorous mathematical foundation for these schemes to ensure robustness and uniform stability for all grid sizes and embedded geometries. Appealing properties of embedding methods such as the ability to model moving interfaces and the introduction of virtual interfaces to enhance parallel performance will be exploited to model a variety of wave problems in electromagnetics, acoustics, seismology, and elasticity. The increasing interest in the accurate and efficient solution of 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 computational methods for such problems. In this effort we propose to develop a new class of computational techniques, specifically aimed at the reliable and robust modeling of problems of a realistic size and complexity, e.g., the modeling of the propagation of electromagnetic and acoustic noise and its environmental impact, and the 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, hence overcoming a number of well known difficulties associated with existing methods which require some kind of automated generation of the computational grid on which the solution is computed. These proposed developments will enable the modeling of very complex and realistic scenarios and will, in combination with high-performance computing facilities for which the methods are well suited, allow for the modeling and analysis of complex wave dominated problems in a variety of areas of interest to engineers and scientists.