This book presents an overview of recent advances in the numerical analysis of nonlinear dispersive partial differential equations (PDEs) - including the nonlinear Schrödinger equation, the Korteweg-de Vries (KdV) equation, and the nonlinear Klein-Gordon equation. These fundamental models are central to mathematical physics and computational PDE theory, and their analysis, both individually and through asymptotic relationships, has become an active and evolving area of research.

Recent progress includes the extension of harmonic analysis tools, such as Strichartz estimates and Bourgain spaces, into discrete settings. These innovations have improved the accuracy and flexibility of numerical methods, especially by relaxing regularity assumptions on initial data, potentials, and nonlinearities. Additionally, enhanced long-time numerical estimates now support simulations over substantially longer time intervals, expanding the practical reach of computational models.

The analytical breakthroughs that underpin these developments trace back to the seminal work by Jean Bourgain in the 1990s, which introduced powerful techniques for studying dispersive PDEs. Adapting these continuous tools to discrete frameworks has proven both challenging and rewarding, offering new insights into the interface between numerical computation and theoretical analysis.

Aimed at graduate students, researchers, and practitioners in numerical analysis, applied mathematics, and computational physics, this volume provides a clear entry point into cutting-edge research, supported by a rich bibliography for further exploration.