| Time and Place: | TuTh 15:30-16:45 Clough UG Lea 423 |
| Reference books: | |
| J. W. Thomas,
Introduction to Numerical Methods for Partial Differential
Equations, Springer, ISBN 0-387-97999-9. |
| L. N. Trefethen,
Spectral Methods in Matlab, SIAM. |
| Claes Johnson, Numerical solution of partial differential equations
by the finite element method. ISBN 0521 345 146. |
| S. C. Brenner and R. Scott,
The Mathematical Theory of Finite Element Methods,
Second Edition. |
| R. J. LeVeque,
Numerical Methods for Conservation Laws,
Birkhauser Verlag, 1992. |
| S. V. Patankar,
Numerical Heat Transfer and Fluid Flow,
Hemisphere Publishing Corp., 1980. |
| C.-W. Shu,
Essentially non-oscillatory and weighted essentially non-oscillatory schemes for hyperbolic conservation laws, in Advanced Numerical Approximation of Nonlinear Hyperbolic Equations, B. Cockburn, C. Johnson, C.-W. Shu and E. Tadmor (Editor: A. Quarteroni), Lecture Notes in Mathematics, volume 1697, Springer, 1998, pp.325-432. |
| |
| Instructor: | Dr. Yingjie Liu |
| Office: | Skiles 134 |
| Phone: | (404)894-2381 |
| E-mail: |
yingjie at math dot gatech dot edu |
| WWW: |
www.math.gatech.edu/~yingjie |
| Office Hours: | TuTh 14:30-15:30 |
| Homework: | Homeworks will be assigned once every couple of
weeks and they must be turned in on time. |
| Grading: | Homeworks 50%, final exam 50%. |
| Topic Outline: | Finite difference methods for parabolic equations, including
heat conduction, forward and backward Euler schemes, Crank-Nicolson
scheme, L infinity stability and L2 stability analysis including Fourier
analysis, boundary condition treatment, Peaceman-Rachford scheme and ADI schemes
in 2D, line-by-line methods etc.
Finite difference and finite volume schemes for hyperbolic equations and
conservation laws, including upwind schemes, Lax-Friedrich scheme, characteristic method,
Lax-Wendroff scheme, MacCormack scheme, back and forth error compensation and correction (BFECC),
time spliting and Strang spliting, Godunov scheme for conservation laws (such as the
Euler equation for gas dynamics), monotone fluxes, MUSCL scheme, ENO scheme, flux corrected transport etc.
Finite element methods for elliptic equations, including
variational formulations, conforming Galerkin methods, Dirichlet boundary condition
treatment and Neumann boundary condition treatment (nature boundary condition),
formation of the stiff matrix, error analysis, steepest decent and conjugate
gradient methods, multigrid method, construction of Lagrangian nodal basis functions etc. |
| |
| Homework 1, due 9/24. |
| Homework 2, due 10/21. |
| Homework 3, due 11/18. |
| Final, due 12/10. |