The application of finite difference schemes in the numerical solution of time-dependent partial differential equations (PDEs) is the topic covered in this book, which is the second edition of a book that was first published in 1989. Some material related to the use of finite difference schemes in the solution of elliptic PDEs is also presented, at the end of the book.

The first five chapters address hyperbolic PDEs and systems of hyperbolic PDEs. The author introduces different types of hyperbolic PDEs (single equations with constant coefficients, systems of equations with constant coefficients, single equations with variable coefficients, systems of equations with variable coefficients, and so on). Both Fourier analysis and Von Neumann analysis of finite difference schemes are presented. Both the order of accuracy and the stability of the finite difference schemes are studied. The description of the use of finite difference schemes in the numerical treatment of hyperbolic PDEs ends with several remarks on the dispersion, dissipation, group velocity, and propagation of wave packets.

Single parabolic PDEs (with constant and variable coefficients) are studied in chapter 6, and systems of PDEs of higher dimensions are discussed in chapter 7. Second-order PDEs are presented in chapter 8. Different properties of finite difference schemes, when these are applied to these types of PDEs, are derived.

Several problems that are related to well-posed and stable problems are discussed in the next three chapters (chapters 9 through 11). Convergence estimates, analysis of the boundary conditions, and stability problems are discussed for several different cases. Elliptic PDEs and different iterative methods used when such PDEs are solved numerically are presented in the last three chapters (chapters 12 through 14). Some additional material (results from matrix and vector analysis, real analysis, and complex analysis) is provided in three appendices at the end of the book. There are 72 references, all of which are papers and books published before 1989. Many examples and exercises are included in the chapters, as well as in the appendices. Careful study of these examples and exercises will be very helpful to readers seeking a better understanding of the material presented in the book.

The book could be used both as a textbook, in courses on PDEs and finite difference schemes, and as a reference book, for researchers working in different fields of science and engineering. In general, it is assumed that readers have had some course in advanced calculus. The results presented in this book could therefore be studied by students with different levels of mathematical background, and could also be used by the scientists in the treatment of different models described by PDEs.