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Introductory textbook from which students can approach more advance topics relating to finite difference methods.
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples. Exercises and student projects are available on the book’s webpage, along with Matlab mfiles for implementing methods. Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics.
Table of contents:
Preface; Part I. Boundary Value Problems and Iterative Methods: 1. Finite difference approximations; 2. Steady states and boundary value problems; 3. Elliptic equations; 4. Iterative methods for sparse linear systems; Part II. Initial Value Problems; 5. The initial value problem for ordinary differential equations; 6. Zero-stability and convergence for initial value problems; 7. Absolute stability for ordinary differential equations; 8. Stiff ordinary differential equations; 9. Diffusion equations and parabolic problems; 10. Advection equations and hyperbolic systems; 11. Mixed equations; A. Measuring errors; B. Polynomial interpolation and orthogonal polynomials; C. Eigenvalues and inner-product norms; D. Matrix powers and exponentials; E. Partial differential equations; Bibliography; Index.
This book introduces finite difference methods for both ordinary differential equations (ODEs) and partial differential equations (PDEs) and discusses the similarities and differences between algorithm design and stability analysis for different types of equations. A unified view of stability theory for ODEs and PDEs is presented, and the interplay between ODE and PDE analysis is stressed. The text emphasizes standard classical methods, but several newer approaches also are introduced and are described in the context of simple motivating examples. Exercises and student projects are available on the book’s webpage, along with Matlab mfiles for implementing methods. Readers will gain an understanding of the essential ideas that underlie the development, analysis, and practical use of finite difference methods as well as the key concepts of stability theory, their relation to one another, and their practical implications. The author provides a foundation from which students can approach more advanced topics.
Table of contents:
Preface; Part I. Boundary Value Problems and Iterative Methods: 1. Finite difference approximations; 2. Steady states and boundary value problems; 3. Elliptic equations; 4. Iterative methods for sparse linear systems; Part II. Initial Value Problems; 5. The initial value problem for ordinary differential equations; 6. Zero-stability and convergence for initial value problems; 7. Absolute stability for ordinary differential equations; 8. Stiff ordinary differential equations; 9. Diffusion equations and parabolic problems; 10. Advection equations and hyperbolic systems; 11. Mixed equations; A. Measuring errors; B. Polynomial interpolation and orthogonal polynomials; C. Eigenvalues and inner-product norms; D. Matrix powers and exponentials; E. Partial differential equations; Bibliography; Index.