- Aims
- Outline
- References
- Prerequisites
- Schedule for Spring 2009
- Marking scheme for Spring 2009
- Announcements for current students

- Introduce the basic concepts in the numerical solution of PDEs.
- Formulate numerical methods for solving PDEs and study their properties.
- Implement the above methods
*efficiently*on the computer. - Use the ELLPACK package for PDEs and packages for sparse linear systems.
- Use MATLAB to study the properties of linear systems arising from the discretisation of PDEs.
- Study the performance of methods and machines.

- Introduction
- Classes of problems and PDEs
- Types of boundary conditions
- Classes of numerical methods for PDEs
- Definitions
- Analysis of numerical methods for PDEs -- error, time, memory

- Boundary value problems: Elliptic PDEs
- Finite difference methods, second order, higher order
- Finite element methods: Galerkin, collocation
- Convergence
- Overview of linear solvers for PDEs

- Initial value problems: Parabolic and Hyperbolic PDEs
- Parabolic PDEs -- Finite difference methods
- Hyperbolic PDEs -- Finite difference methods
- Convergence and Stability
- Method of lines

- Advanced and parallel methods for PDEs (time permitting)
- Parallel methods for BVPs
- Domain decomposition, Schur complement method
- Schwarz alternating (splitting) method
- Multigrid method
- Fast Fourier Transform solvers

General on Partial Differential Equations |
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K. W. Morton and D. F. Mayers | Numerical Solution of Partial Differential Equations | Cambridge University Press 1996 |

Arieh Iserles | A first course in the Numerical Analysis of Differential Equations | Cambridge University Press 1996 |

Michael A. Celia and William G. Gray | Numerical methods for differential equations | Prentice Hall 1992 |

William F. Ames | Numerical Methods for Partial Differential Equations | Academic Press 1992 3rd edition (or 2nd edition) (or Thomas Nelson & Sons) |

C. A. Hall and T. A. Porsching | Numerical Analysis of Partial Differential Equations | Prentice Hall 1990 |

John C. Strikwerda | Finite Difference schemes and Partial Differential Equations | Wadsworth and Brooks/Cole 1989 |

Eric B. Becker, Graham F. Carey and J. Tinsley Oden | Finite Elements (Vol I) | Prentice Hall 1981 |

A. R. Mitchell and R. Wait | The Finite Difference Method in Partial Differential Equations | John Wiley and Sons 1977 |

A. R. Mitchell and R. Wait | The Finite Element Method in Partial Differential Equations | John Wiley and Sons 1977 |

Garrett Birkhoff and Robert E. Lynch | Numerical Solution of Elliptic Problems | SIAM 1984 |

John R. Rice and R. F. Boisvert | Solving Elliptic Problems with ELLPACK | Springer-Verlag, New York 1985 |

Splines and the Finite Element Method |
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P. M. Prenter | Splines and Variational Methods | John Wiley & Sons 1975 |

Carl de Boor | A Practical Guide to Splines | Springer-Verlag 1978 |

Strang and Fix | An Analysis of the Finite Element Method | Prentice Hall |

Martin H. Schultz | Spline Analysis | Prentice Hall 1973 |

Numerical Linear Algebra |
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William W. Hager | Applied Numerical Linear Algebra | Prentice Hall 1988 |

Gene Golub and Charles Van Loan | Matrix computations | John Hopkins University Press 1996 |

General Numerical Analysis |
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S. D. Conte and Carl de Boor | Elementary Numerical Analysis | McGraw-Hill Inc. |

David Kincaid and Ward Cheney | Numerical Analysis | Brooks/Cole 1996 |

Michael Heath | Scientific Computing: an introductory survey | McGraw-Hill Inc. 1997 |

Richard L. Burden and J. Douglas Faires | Numerical Analysis | Brooks/Cole 1997, 6th edition |

L. W. Johnson and R. D. Riess | Numerical Analysis | Addison Wesley |

Advanced and Parallel Methods |
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Yousef Saad | Iterative Methods for Sparse Linear Systems | PWS 1996 |

Wolfgang Hackbusch | Iterative Solution of Large Sparse Systems of Equations | Springer Verlag 1994 |

O. Axelsson and V. A. Barker | Finite element solution of boundary value problems | Academic Press 1984 |

William L. Briggs | A multigrid tutorial | SIAM 1987 |

Gene Golub and Charles Van Loan | Matrix computations | John Hopkins Univ. Press 1996 |

James M. Ortega | Introduction to Parallel and Vector Solution of Linear Systems | Plenum Press 1988 |

Jianping Zhu | Solving Partial Differential Equations on Parallel Computers | World Scientific 1994 |

- Calculus: excellent knowledge and manipulation of Taylor series, differentiation of functions, continuity, limits, Rolle's theorem, mean-value theorem, de l' Hospital's rule, graphs of functions, etc.
- Numerical Linear Algebra (included in CSC350): knowledge of direct and iterative methods for solving linear systems. Fluency in matrix and vector manipulation, both algebraically and algorithmically.
- Interpolation (included in CSC351): approximation of functions by interpolating polynomials, piecewise polynomials, splines, error formulae.
- Numerical integration (included in CSC351): basic numerical integration rules.
- Programming: proficiency in some conventional programming language, such as FORTRAN or C; MATLAB; some minimal knowledge of FORTRAN may be required.

Lectures | Wednesday 1-3 | Room BA B024 |

Friday 2-3 | Room BA B024 | |

Office Hours | Tuesday 4-5 | Room BA 4226 |

** Tentative marking scheme for Spring 2009 **

Problem set 1 | 15% |

Problem set 2 | 15% |

Problem set 3 | 25% |

Problem set 4 | 15% |

Midterm test | 30% |