We propose a boundary method for the numerical simulation of time-dependent waves in three-dimensional (3D) exterior regions. The order of accuracy can be either second or fourth in both space and time. The method reduces a given initial boundary value problem for the wave equation to a set of operator equations at the boundary of the original domain. The reduction is based on a reformulation of the method of difference potentials. The resulting operator equations relate the solution and its normal derivative at the boundary. To solve these equations, one relies on the Huygens' principle. This yields an algorithm that works on a sliding time window of a finite nonincreasing duration. As a result, it allows one to avoid the ever increasing backward dependence of the solution on time. The major advantages of the proposed methodology are its reduced computational complexity (grid-independent on the boundary and sublinear in the volume), the capacity to handle curvilinear geometries using Cartesian finite difference time domain (FDTD) methods, and automatic and exact accounting for the far-field radiation conditions. In addition, the methodology facilitates solution of multiple similar problems al low individual cost per problem and guarantees uniform performance over arbitrarily long time intervals.
- Huygens' principle
- Initial boundary value problem
- Method of difference potentials
- Time-dependent wave (d'Alembert) equation