Numerical Solution of Partial Differential Equations I. Initial value problems for hyperbolic and parabolic differential equations 1. Properly posed initial value problems 2. Types and characteristics 3. Characteristic methods for first order hyperbolic systems 4. Banach spaces 5. Stability of difference methods 6. Examples of stable difference methods 7. Inhomogeneous initial value problems 8. Difference methods with positivity properties 9. Fourier transforms of difference methods 10. Initial value problems in several space variables 11. Extrapolation methods II. Boundary value problems for elliptic differential equations 12. Properly posed boundary value problems 13. Difference methods 14. Variational methods 15. Hermite interpolation and its application to the Ritz method 16. Collocation methods and boundary integral methods III. Solving systems of equations 17. Iterative methods for solving systems of linear and nonlinear equations 18. Overrelaxation methods for systems of linear equations 19. Overrelaxation methods for systems of nonlinear equations 20. Band width reduction for sparse matrices 21. Buneman Algorithm 22. The Schröder Trottenberg reduction method Appendices: Fortran programs Appendix 0: Introduction Appendix 1: Method of Massau Appendix 2: Total implicit difference method for solving a nonlinear parabolic differential equation Appendix 3: Lax Wendroff Richtmyer method for the case of two space variables Appendix 4: Difference methods with SOR for solving the Poisson equation on nonrectangular regions Appendix 5: Programs for band matrices Appendix 6: The Buneman algorithm for solving the Poisson equation. This book is the result of two courses of lectures given at the University of Cologne in Germany in 1974 75. The majority of the students were not familiar with partial differential equations and functional analysis. This explains why Sections 1 2 4 and 12 contain some basic material and results from these areas. The three parts of the book are largely independent of each other and can be read separately. Their topics are: initial value problems boundary value problems solutions of systems of equations. There is much emphasis on theoretical considerations and they are discussed as thoroughly as the algorithms which are presented in full detail and together with the programs. We believe that theoretical and practical applications are equally important for a genuine understa ing of numerical mathematics. When writing this book we had considerable help and many discussions with H. W. Branca R. Esser W. Hackbusch and H. Multhei. H. Lehmann B. Muller H. J. Niemeyer U. Schulte and B. Thomas helped with the completion of the programs and with several numerical calculations. Springer Verlag showed a lot of patience and under­ standing during the course of the production of the book. We would like to use the occasion of this preface to express our thanks to all those who assisted in our sometimes arduous task.