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.mfun_Newton.mdfun_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
Matlab code for Trapezoidal rule: trapezoidal.m, fun.m, dfun.m
  •  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
Matlab script of Five-point formula, Nine-point formula
  • 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.

Assignments
   
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