CSC446-2310S Computational Methods for Partial Differential Equations

• 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

References

General on Partial Differential Equations
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
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
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
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
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

Prerequisites

• 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.

Schedule for Spring 2009

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%

Most of the assignments include substantial computer work.