This book, "Numerical Partial Differential Equations: Finite Difference Methods" by J.W. Thomas, is a comprehensive textbook for graduate students in applied mathematics and engineering. It provides an in-depth exploration of numerical methods for solving partial differential equations (PDEs), with a focus on finite difference methods. The book is divided into two parts: the first part covers finite difference methods for time-dependent equations, including parabolic and hyperbolic problems, multi-dimensional problems, systems, and dissipation and dispersion. The second part discusses stability theory for initial-boundary value problems, numerical schemes for conservation laws, numerical solution of elliptic problems, and an introduction to irregular regions and grids. The book emphasizes both the implementation of numerical schemes and the theoretical foundations of PDEs. It includes detailed discussions on convergence, consistency, and stability, as well as the Lax Theorem. The text also covers various numerical methods, including implicit and explicit schemes, and discusses the Courant-Friedrichs-Lewy condition. It addresses the numerical solution of conservation laws, including shock formation, weak solutions, and entropy conditions. The book also explores elliptic equations, including solvability, convergence, and residual correction methods, as well as the use of iterative methods and fast Fourier transforms for solving time-dependent problems. The text includes numerous computational interludes that provide practical examples and exercises for students to apply the concepts learned. It also discusses the derivation of numerical schemes, both mathematically and based on physical principles, and emphasizes the importance of understanding numerical integration errors. The book is designed to be a reference for students and professionals in applied mathematics and engineering, offering a thorough understanding of numerical methods for solving PDEs.This book, "Numerical Partial Differential Equations: Finite Difference Methods" by J.W. Thomas, is a comprehensive textbook for graduate students in applied mathematics and engineering. It provides an in-depth exploration of numerical methods for solving partial differential equations (PDEs), with a focus on finite difference methods. The book is divided into two parts: the first part covers finite difference methods for time-dependent equations, including parabolic and hyperbolic problems, multi-dimensional problems, systems, and dissipation and dispersion. The second part discusses stability theory for initial-boundary value problems, numerical schemes for conservation laws, numerical solution of elliptic problems, and an introduction to irregular regions and grids. The book emphasizes both the implementation of numerical schemes and the theoretical foundations of PDEs. It includes detailed discussions on convergence, consistency, and stability, as well as the Lax Theorem. The text also covers various numerical methods, including implicit and explicit schemes, and discusses the Courant-Friedrichs-Lewy condition. It addresses the numerical solution of conservation laws, including shock formation, weak solutions, and entropy conditions. The book also explores elliptic equations, including solvability, convergence, and residual correction methods, as well as the use of iterative methods and fast Fourier transforms for solving time-dependent problems. The text includes numerous computational interludes that provide practical examples and exercises for students to apply the concepts learned. It also discusses the derivation of numerical schemes, both mathematically and based on physical principles, and emphasizes the importance of understanding numerical integration errors. The book is designed to be a reference for students and professionals in applied mathematics and engineering, offering a thorough understanding of numerical methods for solving PDEs.