Readings are assigned in the following textbooks:

[Evans] = Evans, Lawrence C. Vol. 19. Graduate studies in mathematics. Providence, RI: American Mathematical Society, 1998. ISBN: 9780821807729.

[Strang] = Strang, Gilbert. *Computational Science and Engineering*. Wellesley, MA: Wellesley-Cambridge Press, 2007. ISBN: 9780961408817.

[LeVeque2007] = LeVeque, Randall J. *Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems*. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2007. ISBN: 9780898716290.

[Trefethen] = Trefethen, Lloyd N. *Spectral Methods in MATLAB (Software, Environments, Tools)*. Philadelphia, PA: Society for Industrial and Applied Mathematics, 2001. ISBN: 9780898714654.

[LeVeque2002] = LeVeque, Randall J. *Finite Volume Methods for Hyperbolic Problems*. Cambridge texts in applied mathematics. Cambridge, UK: Cambridge University Press, 2002. ISBN: 9780521009249.

[Fletcher] = Fletcher, C. A. J. *Computational Techniques for Fluid Dynamics. Fundamental and General Techniques Volume I*. Springer series in computational physics. New York, NY: Springer-Verlag, 1996. ISBN: 9783540530589.

[Canuto] = Canuto, Claudio S., M. Y. Hussaini, A. Quarteroni, and T. A. Zang. *Spectral Methods Evolution to Complex Geometries and Applications to Fluid Dynamics*. New York, NY: Springer-Verlag, 2007. ISBN: 9783540307273.

SES # | TOPICS | READINGS |
---|---|---|

1 | Fundamental concepts and examples | Evans 1.1, 1.2 |

2 | Well-posedness and Fourier methods for linear initial value problems |
Evans 1.3 Strang 6.1 |

3 | Laplace and Poisson equation | Evans 2.2 |

4 | Heat equation, transport equation, wave equation | Evans 2.1 and 2.3-2.4 |

5 | General finite difference approach and Poisson equation | LeVeque2007, chapter 1, 2.1-2.4, 2.12 |

6 | Elliptic equations and errors, stability, Lax equivalence theorem | LeVeque2007 2.5-2.9, 2.15 (recommended 2.17) |

7 | Spectral methods | Trefethen, chapters 1, 3 |

8 | Fast Fourier transform (guest lecture by Stephen Johnson) | |

9 | Spectral methods | Trefethen, chapters 6, 7, 8 |

10 | Elliptic equations and linear systems | LeVeque2007, chapter 3 |

11 | Efficient methods for sparse linear systems: multigrid |
LeVeque2007 4.1-4.2, 4.6 Strang 7.1 |

12 | Efficient methods for sparse linear systems: Krylov methods | LeVeque2007 4.3-4.4 |

13 | Ordinary differential equations | LeVeque2007 9.2, chapter 5 |

14 | Stability for ODE and von Neumann stability analysis | LeVeque2007, chapter 7, 9.6-9.7 |

15 | Advection equation and modified equation | LeVeque2002 4.4-4.9, 8.6 |

16 | Advection equation and ENO/WENO | LeVeque2002 6.1, 6.3, 6.7, 10.4 |

17 | Conservation laws: theory | LeVeque2002, chapter 11 |

18 | Conservation laws: numerical methods | LeVeque2002 12.1-12.9, (recommended 12.10-12.11) |

19 | Conservation laws: high resolution methods | LeVeque2002 12.12, 6.4-6.12 |

20 | Operator splitting, fractional steps | |

21 | Systems of IVP, wave equation, leapfrog, staggered grids |
Notes on the wave equation (PDF) Notes on perfectly matching layers (PDF) (Courtesy of Steven G. Johnson. Used with permission.) |

22 | Level set method |
Notes on the level set method (PDF) Presentation on the level set method (PDF) (Courtesy of Per Olof Persson. Used with permission.) |

23 | Navier-Stokes equation: finite difference methods |
Fletcher, Canuto |

24 | Navier-Stokes equation: pseudospectral methods |
Fletcher, Canuto (no spefic section) |

25 | Particle methods | |

26 | Project presentations |