Numerical Solutions of Differential Equations

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 (qigong@soe.ucsc.edu), Baskin Engineering 361A

Text Book

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

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, 4:00PM - 5:45PM 

Office Hours: Wednesday 12:30PM - 2:30PM, BE 361A

Grading: Homework 50%, Final exam 50%.

Tentative Schedule

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.

Instructors and Assistants