chapter 7, Numerical Methods for Nonlinear Wave Equations

Excerpt

Numerical computations play an important role in the study of nonlinear waves. Many important phenomena, such as elastic collision of KdV solitons and fractal scattering in solitary wave interactions, were discovered first by numerical computations (see Zabusky and Kruskal (1965) and Chapter 5). Such numerical discoveries motivated further analytical investigations, which then led to deeper understanding of those phenomena. Numerical computations are especially important for the study of nonintegrable equations. Indeed, in Chapters 5 and 6 on nonintegrable equations, many numerical results were presented. A dynamic interplay between numerics and analysis was a distinctive feature for the topics in those chapters. In recent years, great advances have been made on numerical methods for nonlinear wave equations, and many highly accurate and efficient numerical techniques have been developed. In addition, some convenient commercial software for scientific computing (such as MATLAB) have also become available, which greatly facilitated the coding and implementation of those numerical algorithms. Due to these advances, nontrivial numerical computations of nonlinear wave equations can often be done with several-inch-long MATLAB codes and hence are accessible to anybody with or without prior programming experience.

In this chapter, we describe numerical methods for various aspects of nonlinear wave computations such as evolution simulation, computation of solitary wave solutions, and computation of linear-stability spectra of solitary waves. Fundamental properties of these methods, such as their accuracy, numerical stability, convergence conditions, and convergence speeds, will be elaborated. We will show that all these methods we present have spectral spatial accuracy; i.e., the spatial error decays exponentially with the grid spacing (for smooth functions). Thus these methods yield highly accurate numerical results.