AMS 213B, Spring 2016

Numerical Methods for the Solution of Differential equations


 

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

Office: Baskin Engineering 353C

Office Hours: Monday 1:30 pm - 2:30 pm (or by appointment)

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

  


 

Course Objectives

This graduate level course provides an introduction to the numerical solutions of ordinary and partial differential equations (ODEs and PDEs). The course focuses on the derivation of discrete solution methods for a wide variety of differential equations and their stability and convergence.  The course also provides hands-on experience on implementing numerical algorithms for solving engineering and scientific problems using the widely available MATLAB software.  Class will be taught by classroom lectures and hands-on programming sections.

 


 

Programming Language

We will use Matlab for the course.

 


  

Course Materials

1. Lecture note (main),

2. Selected Books:

General

  • “Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations’’ by L. N. Trefethen (available online at http://people.maths.ox.ac.uk/trefethen/pdetext.html)
  • “Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems” by R. LeVeque, ISBN 0898716292
  • “Numerical Methods for Wave Equations in Geophysical Fluid Dynamics” by Dale R. Durran, ISBN 0-387-98376-7
  • “Numerical Recipes: The Art of Scientific Computing” by W.H. Press, S.A. Teukolsky, W.T. Vetterling & B.P. Flannery, ISBN 0521880688 available online at http://www.nr.com/oldverswitcher.html

ODEs

  • “Numerical Solution of Ordinary Differential Equations” by K. Atkinson, W. Han & D.E. Stewart, ISBN 047004294X

  • “Numerical Methods for Ordinary Differential Equations” by J. Butcher, ISBN 0470723351
  • “Scientific Computing and Differential Equations: An Introduction to Numerical Methods” by G. Golub & J. Ortega, ISBN 0122892550

PDEs

  • “Numerical Solution of Partial Differential Equations” by K.W. Morton & D.F. Mayers, ISBN 0521607930; “Numerical Solution of Partial Differential Equations: Finite Difference Methods” by G.D. Smith, ISBN 0198596502

  • “Numerical Partial Differential Equations: Finite Difference Methods” by J.W. Thomas, ISBN 0387979999

  • “Implementing Spectral Methods for Partial Differential Equations: Algorithms for Scientists and Engineers” by D. Kopriva, ISBN 9048122600

  • “Chebyshev and Fourier Spectral Methods” by J. Boyd, available online at http://www-personal.umich.edu/~jpboyd/BOOK_Spectral2000.html

MATLAB Specific

  • “Spectral Methods in MATLAB” by L. N. Trefethen, ISBN 0898714656
  • “Applied Numerical Methods using MATLAB” by Laurene Fausett ISBN 0132397385
  • “A First Course in Computational Physics” by P. DeVries & J. Hasbun, ISBN 9780763773144
  • “Applied Numerical Analysis” by C. Gerald & P. Wheatley, ISBN 0321133048

 


  

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 (50%): May 25, 2016
  • Final-term computer programming project (20%): Due June 8, 2016 (tentative) -- In a final project you will be asked to implement numerical schemes to solve a physics problem. It is also required to write a scientific report in a professional style using LaTex 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 20% of your total grade.

 

 


 

Syllabus

  • Week 1: Motivation and introduction to differential equations and concepts of accuracy and stability. ODEs: 1-step Euler methods: derivation; explicit; implicit; accuracy and stability; trapezoidal rule. Methods of derivation: Taylor series; undetermined coefficients; polynomial fitting.
  • Week 2: ODEs: Multistep methods: Adams-Bashforth; Adams-Moulton. Stability: zero-stability; A-stability. Convergence. Lax Equivalence Theorem.
  • Week 3: ODEs: Compound methods: Predictor-Corrector; Runge-Kutta. Extrapolation methods: Richardson; Gragg; Bulirsch-Stoer. Systems of equations: stiffness.
  • Week 4: ODEs: Two-point boundary-value problems: shooting; relaxation; projection.
  • Week 5: PDEs: Classification. General concepts of finite-difference approximations. Methods for parabolic equations: Forward-Time Centered-Space (FTCS; explicit); Backward-Time Centered-Space (BTCS: implicit); Crank-Nicolson. Von Neumann stability analysis.
  • Week 6: PDEs: Methods for hyperbolic equations: Failure of FTCS; Lax method. Courant-Friedrich-Lewy (CFL) condition. Other types of errors: phase (dispersion errors); nonlinear (numerical diffusion); transport errors (leading to upwind schemes).
  • Week 7: PDEs: Methods for hyperbolic equations (continued): Second order methods: Leapfrog; 2 step Lax-Wendroff. Higher order finite-difference methods: 3rd order Essentially Non-Oscillatory; 5th order Weighted ENO (WENO).
  • Week 8: PDEs: Methods for elliptic equations: 5-point finite-difference stencil. Review of direct solvers: Gaussian elimination; LU. Review of iterative solvers: Jacobi iteration; Gauss-Seidel iteration; Successive Over-Relaxation; Conjugate Gradient; preconditioning; accelerators (red-black coloring; multigrid). Dirichlet/Neumann boundary conditions.
  • Week 9: Multidimensional problems: Alternating Direction Implicit (ADI). Other rapid elliptic solvers: Spectral methods, including Fourier; Chebyshev; pseudo-spectral methods; FFTs; dealiasing; difference between Galerkin, tau, collocation methods.
  • Week 10: A glimpse at other methods (at discretion of instructor): Finite element, finite volume, etc.

 


 

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.