Numerical Solutions of Partial Differential Equations

ISBN-13: 9780387905501 / Angielski / Miękka / 1981 / 556 str.

Theodor Meis; T. Meis; U. Marcowitz

Opis

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.

Spis treści

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.