AMS 213 – Numerical Solutions of Differential Equations. The course focuses on basic numerical linear algebra and numerical methods for ODEs and PDEs. Topics include LU, Cholesky and QR decompositions; iterative methods; Runge-Kutta methods; error estimation and error control; stability and convergence, etc. MATLAB is the chosen language for this course.
Instructor: Qi Gong (email@example.com), Baskin Engineering 361A
The lectures are based on notes extracted from various textbooks instead of using a single textbook. Listed in the following are some suggested references.
Numerical Linear Algebra, Trefethen & Bau, SIAM
Numerical Analysis of Differential Equations, Arieh Iserles, Cambridge Univ. Press
Numerical Recipes, Teukolsky, Vetterling & Flannery, Cambridge Univ. Press
Mastering MATLAB 6, Duane Hanselman and Bruce Littlefield, prentice Hall
Introduction to MATLAB, Ross L. Spencer and Michael Ware, available at http://www.physics.byu.edu/Courses/Computational/phys330/matlab.pdf
Lectures: Tuesday and Thursday, 10:00AM - 11:45AM, Crown Cirm 104
Office Hours: Monday 9:00AM - 11:30AM, BE 361A
Grading: Homework 50%, Final exam 50%.
Week 1: Introduction, gaussian elimination, LU factorization and Cholesky factorization.
Week 2: Mathematical preliminaries on linear space, QR factorization by Gram-Schmidt and Householder transformation.
Week 3: Lest squares fitting, condition number and stability.
Week 4: Iterative methods for linear systems, conjugate gradient method.
Week 5: Eigenvalue and eigenvector problem.
Week 6: Initial value problem of ODEs. Eluer’s method, Explicit Runge-Kutta methods, convergence rate analysis, stiff equations and absolute stability.
Week 7: Quadrature integration and Implicit Runge-Kutta and multistep methods.
Week 8: Two point boundary value problems, finite difference and shooting methods. PDE 1: Poisson equations.
Week 9: PDE 2: Diffusion equations, consistency, order and stability. Lax Equivalence Theorem.
Week 10: PDE 3: Stability analysis: eigenvalue approach and Fourier transform approach, advection equations.