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.