Numerical Methods for Fluids, Part 3

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Front Cover 2
Numerical Methods for Fluids (Part 3) 5
Copyright Page 6
Contents 17
Preface 21
Chapter I. The Navier–Stokes Equations for Incompressible Viscous Fluids 25
- Introduction: Synopsis 25
- 1. Derivation of the Navier–Stokes equations for viscous fluids 25
- 2. Initial and boundary conditions 33
- 3. A stream function-vorticity formulation of the Navier–Stokes equations 35
- 4. A brief introduction to Sobolev spaces 39
- 5. Variational formulations of the Navier–Stokes equations 46
- 6. A short review of mathematical results for the Navier–Stokes equations 60
Chapter II. A Family of Operator-Splitting Methods for Initial Value Problems. Application to the Navier–Stokes Equationsh 63
- Introduction: Synopsis 63

Front Cover 2
Numerical Methods for Fluids (Part 3) 5
Copyright Page 6
Contents 17
Preface 21
Chapter I. The Navier–Stokes Equations for Incompressible Viscous Fluids 25
- Introduction: Synopsis 25
- 1. Derivation of the Navier–Stokes equations for viscous fluids 25
- 2. Initial and boundary conditions 33
- 3. A stream function-vorticity formulation of the Navier–Stokes equations 35
- 4. A brief introduction to Sobolev spaces 39
- 5. Variational formulations of the Navier–Stokes equations 46
- 6. A short review of mathematical results for the Navier–Stokes equations 60
Chapter II. A Family of Operator-Splitting Methods for Initial Value Problems. Application to the Navier–Stokes Equationsh 63
- Introduction: Synopsis 63
- 8. The Peaceman–Rachford method 64
- 7. A family of initial value problems 64
- 9. The Douglas–Rachford method 72
- 10. A θ -scheme 75
- 11. Application to the Navier–Stokes equations 83
- 12. Further comments 85
Chapter III. Iterative Solution of the Advection-Diffusion Subproblems 89
- Introduction: Synopsis 89
- 13. Classical and variational formulations of the advection-diffusion subproblems associated with the operator splitting schemes 90
- 14. Linear variational problems in Hilbert spaces 92
- 15. Variational methods for the solution of the advection-diffusion problems (13.1) and (13.2) 114
- 16. Conjugate gradient methods for the solution of minimization problems in Hilbert spaces 135
- 17. Least squares solution of linear and nonlinear problems in Hilbert spaces 155
- 18. Least-squares/conjugate gradient solution of problems (13.1) and (13.2) 193
Chapter IV. Iterative Solution of the Stokes Subproblems 201
- Introduction: Synopsis 201
- 19. Mathematical properties of the generalized Stokes problem (GS)1 202
- 20. Gradient methods for the Stokes problem 230
- 21. Conjugate gradient methods for the Stokes problem (GS)1 259
- 22. Iterative solution of the generalized Stokes problem (GS)2 272
- 23. On artificial compressibility methods and further comments 280
Chapter V. Finite Element Approximation of the Navier–Stokes Equations 305
- Introduction: Synopsis 305
- 24. Solution of the Stokes problem with periodic boundary conditions 307
- 25. A Fourier analysis of the numerical instability mechanism 309
- 26. Finite element methods for the Stokes problem 314
- 27. Finite element implementation of the θ -scheme (11.5)–(11.8) 396
- 28. On the numerical solution of the discrete subproblems 424
- 29. Further comments and complements 431
Chapter VI. Treatment of the Advection by a Wave-Like Equation Method and by Backward Methods of Characteristics 445
- Introduction: Synopsis 445
- 30. More on operator-splitting methods 446
- 31. A wave-like equation method for solving the Navier–Stokes equations 510
- 32. Solution of the Navier–Stokes equations by backward methods of characteristics 553
- 33. On the treatment of the advection by upwinding. Final comments 567
Chapter VII. On L2-Projection Methods for the Numerical Treatment of the Incompressibility 577
- Introduction: Synopsis 577
- 34. Combining L2-projection methods with operator-splitting schemes à la Peaceman–Rachford and Douglas–Rachford, and with the th 578
- 35. Combining L2-projection methods with operator-splitting schemes à la Marchuk–Yanenko 599
- 36. Numerical experiments 610
- 37. Further comments and references 624
Chapter VIII. Fictitious Domain Methods for Incompressible Viscous Flow: Application to Particulate Flow 631
- Introduction: Synopsis 631
- 38. Generalities on fictitious domain methods 632
- 39. On the solution of Dirichlet problems by fictitious domain methods with boundary supported Lagrange multipliers. Application 634
- 40. A boundary supported Lagrange multiplier/fictitious domain method for the incompressible Navier–Stokes equations 690
- 41. On a fictitious domain method with volume distributed Lagrange multipliers for Dirichlet problems 703
- 42. On the direct numerical simulation of incompressible viscous flow with moving rigid boundary by distributed Lagrange multipl 713
Chapter IX. Numerical Experiments 783
- Introduction: Synopsis 783
- 43. Flow in a nozzle at high incidence 784
- 44. Application of the wave-like equation method to the numerical simulation of incompressible viscous fluid flow in square and 798
- 45. Numerical simulation of incompressible viscous flow a two-dimensional channel with a backward facing step 827
- 46. Numerical simulation of a thermal convection flow in a differentially-heated rectangular cavity 848
- 47. More on particulate flow 866
- 48. On blood flow in the heart 885
Chapter X. Complements: From Stream Function-Vorticity to Flow Control 889
- Introduction: Synopsis 889
- 49. Numerical methods for the stream function-vorticity formulation of the Navier–Stokes equations 890
- 50. Simulation of Bingham visco-plastic flow 952
- 51. On the numerical simulation of slightly compressible isentropic viscous flow 972
- 52. Modeling and simulation of low-Mach-number compressible flows 983
- 53. Optimal control of systems modeled by the incompressibleNavier–Stokes equations: Drag reduction by active control for flow p 1001
Acknowledgements 1061
References 1065
Author Index 1087
Subject Index 1095

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