Numerical Methods II
Spring 2007, Mondays 5:10-7:00 pm, WWH Room 101 Syllabus Lectures - Lecture 1 (Jan 22): Newton's method for system of nonlinear equations, rate of convergence; Broyden's method; Forward (explicit) Euler's method for ODEs, local (or truncation) error, convergence, accuracy.
Matlab code for Newton's method,
and a test example: newton.m, fun_Newton.m, dfun_Newton.m
Forward Euler's method: feuler.m, fun.m - Lecture 2 (Jan 29): Trapezoidal rule; Multistep
methods - Adams Bashforh methods, algebraic condition for the order
- Lecture 3 (Feb 5): Convergence of multistep
methods - consistency and root condition; Construction of
multistep methods; BDF family; Introduction to Runge-Kuta methods
Matlab code for explicit two-step
methods: twostep.m
- Lecture 4 (Feb 12): Derivation/Construction of 1, 2, and 3-stage ERK method by Taylor's expansion; Gaussian quadrature; Collocation RK method
- Lecture 5 (Feb 26): Accuracy of collocation RK methods; Linear stability analysis; Stability region of ERK method; A-stability of Gauss-Legendre RK method
- Lecture 6 (Mar 5): Stability region of multistep methods;
A(\alpha)-stablility of BDF; Variable time step based on error
etimation and control for one-step methods.
Euler's method with variable time
step feuler_varh.m (The modified
Trapezoidal rule is used as the error controller)
- Lecture 7 (Mar 19): Finite difference for two point boundary value problem; Five-point formula for Poisson equation, structure of the matrix; LU factorization of banded matrix
- Lecture 8 (Mar 26): Nine-point formula for Poisson equation; Weak form of two point BVP; Formulation of Galerkin method; Accuracy of Galerkin method: Céa's lemma
- Lecture 9 (April 2): Finite element method
- Lecture 10 (April 9): Finite difference methods for diffusion equations; Lax equivalence theorem; Von Neumann stability analysis.
- Lecture 11 (April 23): Finite difference methods for advection equation; Domain of dependence, CFL condition; Stability analysis.
- Lecture 12 (April 30): Modified equations; Dissipation and
dispersion; Finite difference for the wave equation.
Assignment
1 - Assigned Jan 29, due Feb 19
Assignment 2 - Assigned Feb 26, due Mar 19 Assignment 3 - Assigned March 19, due April 2 Assignment 4 - Assigned April 2, due April 16 Assignment 5 - Assigned April 23, due May 4 |