9. Boundary Modeling and High-Order Convergence in Finite-Difference Methods

High-order finite-difference methods are appealing for large-scale numerical computations, as their excellent numerical dispersion properties enable the use of coarser grids for the modeling of uniform media. However, practical problems of interest involve, in addition to uniform media, complex boundary conditions, including curved boundaries. In fact, the lack of robust methods to incorporate curved material interfaces with consistent error performance is widely considered as a significant bottleneck in the application of high-order finite-difference techniques to practical problems. The present chapter addresses this problem, revisiting the generation of conformal, high-order finite-difference methods from the perspective of transformation electromagnetics. Fundamentally based on the metric invariance property of Maxwell’s equations, transformation electromagnetics and optics has recently been employed in the design of various cloaking media, yet it presents interesting numerical applications as well. After a brief presentation of transformation-driven numerical methods, the consistent, high-order modeling of 2/3-D curved boundaries is discussed.