Finite difference method

Abstract: Solving efficiently the wave equations involved in modeling acoustic, elastic or electromagnetic wave propagation remains a challenge both for research and industry. To attack the problems coming from the propagative character of the solution, the author constructs higher-order numerical methods to reduce the size of the meshes, and consequently the time and space stepping, dramatically improving storage and computing times. This book surveys higher-order finite difference methods and develops various mass-lumped finite (also called spectral) element methods for the transient wave equations, and presents the most efficient methods, respecting both accuracy and stability for each sort of problem. A central role is played by the notion of the dispersion relation for analyzing the methods. The last chapter is devoted to unbounded domains which are modeled using perfectly matched layer (PML) techniques. Numerical examples are given.

Abstract: A family of inexpensive discretization schemes for diffusion problems on unstructured polygonal and polyhedral meshes is introduced. The material properties are described by a full tensor. The theoretical results are confirmed with numerical experiments.

Abstract: We previously introduced the alternating direction implicit finite-difference time domain (ADI-FDTD) method for a two-dimensional TE wave. We analytically and numerically verified that the algorithm of the method is unconditionally stable and free from the Courant-Friedrich-Levy condition restraint. In this paper, we extend this approach to a full three-dimensional (3-D) wave. Numerical formulations of the 3-D ADI-FDTD method are presented and simulation results are compared to those using the conventional 3-D finite-difference time-domain (FDTD) method. We numerically verify that the 3-D ADI-FDTD method is also unconditionally stable and it is more efficient than the conventional 3-D FDTD method in terms of the central processing unit time if the size of the local minimum cell in the computational domain is much smaller than the other cells and the wavelength.