High Order Numerical Simulation of Waves Using Regular Grids and Non-conforming Interfaces.

摘要

We study the propagation of waves over large regions of space with smooth, but not necessarily constant, material characteristics, separated into sub-domains by interfaces of arbitrary shape. We consider a divide and conquer approach based on wave splitting into incoming and outgoing waves. We assemble the overall solution from the set of individual solutions to an auxiliary problem (AP). The AP is defined independently for each sub-domain. The choice of the AP is relatively flexible; it can be formulated to enable an easy and economical numerical solution. Our new method uses only simple structured grids, e.g., Cartesian or polar, regardless of the shape of the boundaries or interfaces. In the regions of smoothness, it employs high order accurate finite difference schemes on compact stencils. They do not require any additional boundary conditions besides those needed for the underlying differential equation itself. Interfaces not aligned with the grid handled by Calderon's operators and the method of difference potentials. The method of difference potentials offers a number of important benefits for computation. It easily handles curvilinear boundaries, variable coeffcients and general boundary conditions while the complexity remains close to that of a finite difference scheme on a regular structured grid. The main advantage is that this methodology provides high order accuracy and overcomes the difficulties inherent in more traditional approaches.