Numerical Solutions of Differential Equations

 

AMS 213: Numerical Solutions of Differential Equations


 

Instructor: Dongwook Lee (dlee79 _at_ ucsc.edu), Applied Mathematics and Statistics

Office: Baskin Engineering 353C

Office Hours: Monday 2 pm - 3 pm (or by appointment)

Lectures: Mon, Wed, & Fri 11:00 am - 12:10 pm at Jack Baskin Engineering classroom 169

 


 

 Course Objectives

This course aims to focus on the basic numerical methods which are most fundamental in studying various fields of scientific computing. The subject of this course is traditionally called numerical analysis which is concerned with the design and analysis of numerical (discrete) algorithms for solving mathematical problems using computers. We achieve our goal by studying several key topics in two parts: (1) numerical linear algebra and (2) numerical algorithms for ODEs and PDEs.

 


 

 Course Materials

Major reading materials: Class lecture notes

Other supplementary reading materials:

  • An Introduction to Numerical Analysis -- Kendall E. Atkinson (Wiley)
  • Scientific Computing, An Introductory Survey -- Michael T. Heath (McGraw Hill)
  • Numerical Linear Algebra -- Lloyd N. Trefethen and David Bau, III (SIAM)
  • Numerical recipes -- Press, Teukolsky, Vetterling and Fannery (Cambridge Univ. Press)
  • Finite Difference Methods for Ordinary and Partial Differential Equations -- Randall J. LeVeque (SIAM)
  • A First Course in the Numerical Analysis of Differential Equations -- Arieh Iserles (Cambridge Univ. Press)
  • High Performance Scientific Computing Online Lecture Note -- Randall J. LeVeque (http://faculty.washington.edu/rjl/classes/am583s2014/notes)

  


 

 Grading Policy

Homework (30%): There are total of 6 homework problem sets on both mathematical theories and computer programming in every week. There is a policy on any late homework submission -- you are going receive a maximum of 80% if late by less than a day; 50% if late by more than a day. Students are strongly encouraged to submit their homework electronically in pdf (no word documents).

Mid-term take-home exam (30%): May 4, 2015 (tentative)

Final-term computer programming project (40%): Due June 11, 2015 (tentative) -- In a final project you will be asked to implement numerical schemes in Fortran 90 to solve a physics problem. It is also required to write a scientific report in a professional style using either latex or any word documents and submit as a pdf file. Please keep in mind that the quality of the project goes past the homework set materials. Project submission is to be made to your git repository by the due date. The project will take 40% of your total grade.

 


 

 A list of course topics (tentative)

  • Week 1: Review of basic ideas in scientific computing, Unix/Linux, introduction to scientific languages, Fortran90, version control using Subversion and Git, review of basic linear algebra
  • Week 2: Direct methods of solving linear systems (Gaussian elimination, LU factorization, Choleski factorization), least square problems (QR factorization)
  • Week 3: Continuing least square problems (orthogonalization methods: Gram-Schmidt, Householder transformations)
  • Week 4: Eigenvalue problems (power iteration, Rayleigh quotient iteration, QR iteration), singular value decomposition and its applications
  • Week 5: Initial value problems for ODEs (single-step and multi-step methods, accuracy and stability, explicit vs. implicit methods)
  • Week 6: (Two-point) Boundary value problems for ODEs (shooting method, finite difference method, Galerkin method)
  • Week 7: Numerical methods for parabolic PDEs (explicit vs. implicit, stability analysis)
  • Week 8: Numerical methods for hyperbolic PDEs (linear advection equations vs. nonlinear Burgers' equation, stability analysis, the Courant condition, the Lax equivalence theorem)
  • Week 9: Continuing Numerical methods for hyperbolic PDEs
  • Week 10: Iterative methods for elliptic PDEs (Jacobi, Gauss-Seidel, SOR, CG)

 



Students with disabilities: 

If you qualify for classroom accommodations because of a disability, please get an Accommodation Authorization from the Disability Resource Center (DRC) and submit it to me in person outside of class (e.g., office hours) within the first two weeks of the quarter. Contact DRC at 459-2089 (voice), 459-4806 (TTY), or http://drc.ucsc.edu for more information on the requirements and/or process.

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