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Preliminaries: 1. Introduction 2. Governing equations Part I. Finite Difference Methods: 3. Derivation of finite difference equations 4. Solution methods of finite difference equations 5. Incompressible viscous flows via FDM 6. Compressible flows via FDM 7. Finite volume methods via FDM Part II. Finite Element Methods: 8. Introduction to finite element methods 9. Finite element interpolation functions 10. Linear problems 11. Nonlinear problems 12. Incompressible viscous flows via FEM 13. Compressible flows via FEM 14. Miscellaneous weighted residual methods 15. Finite volume methods via FEM 16. Relationships between finite differences and finite elements and other methods Part III. Automatic Grid Generation, Adaptive Methods and Computing Techniques: 17. Structured grid generation 18. Unstructured grid generation 19. Adaptive methods 20. Computing techniques Part IV. Applications: 21. Applications to turbulence 22. Applications to chemically reactive flows and combustion 23. Applications to acoustics 24. Applications to combined mode radiative heat transfer 25. Applications to multiphase flows 26. Applications to electromagnetic flows 27. Applications to relativistic astrophysical flows Appendices.