Textbook for Part II: Spectral Methods in Matlab by L. N. Trefethen, SIAM, PA (2000). ISBN 0-89871-465-6.

Course Outline

The course consists of two parts as indicated above by the textbook description. The textbook for the first part is the same as the one we used in course MATH712. The second book (paperback and cheap!) will be used for the second part of the course. Notes, papers, and other material will be available on reserve in the Library.

In part I we will extend the material presented in MATH712 to derive and analyze finite difference schemes for two dimensional systems of partial differential equations of physical interest. Issues pertaining to the implementation and stability analysis of physical and numerical boundary conditions will be discussed. Two applications will be considered in detail: a) the time-dependent Maxwell equations of electrodynamics in two spatial dimensions, b) the unsteady incompressible Navier-Stokes equations of flyid mechanics in two spatial dimensions.

In part II we will discuss pseudo-spectral and spectral methods to solve partial differential equations. In these methods the solution is expanded as a Fourier series or as a series in Chebyshev polynomials. We will discuss the approximation properties of Fourier and Chebyshev series, along with techniques based on the Fast Fourier Transform (FFT) and on matrix multiplication to numerically compute partial derivatives. Time-discretization techniques suitable for use with pseudo-spectral and spectral methods will be analyzed. We will apply such methods to solve systems of partial differential equations in one spatial dimension.

The following chapters of the textbooks will accompany us in the journey:

Part I: Chapter 4: Sections 4.2, 4.3, 4.4. Chapter 5: Section 5.8. Chapter 6: Sections 6.7, 6.8. Chapter 7: This chapter will be adapted to two-dimensional PDE's. From NOTES: Stability of two-dimensional initial boundary value problems using finite difference schemes. Applications: See description above.

Part II: Chapters 1-8, 10. Notes on implementation and analysis of spectral methods.

Exams

There will be no exams in this course. Students will be evaluated on the basis of regular homework (that will include computational components) and a final project.

Homework/Computational Projects

Homework will be assigned regularly. Some (or all) of the homework problems in each assignment will require computer programming, use of graphics packages for the presentation of data obtained from the computer program. You are free to use any programming language you like, but note I can only help you with Fortran and Matlab. There are many graphics packages on our systems, if you can't pick one yourself ask me for suggestions. There will be one major computational project, assigned two weeks before the last day of classes and collected on the day of the scheduled final exam. A completed project will be in the form of a research paper (details of the structure to be given later). Your ability to complete the project will depend on your having done all the programming assignments in the homework sets.

Grading

The homework assignments will account for 60% of the course grade. The final computational project will account for the remaining 40% of the course grade.

Library Reserve Materials

A copy of the book Finite Difference Schemes and Partial Differential Equations (by J. C. Strikwerda) has been placed on reserve for in-library use by students of MATH707. Other course-related materials will be placed on reserve at later dates (course notes, etc.).

Course Web Page

The course web page (always under construction !) at http://web.njit.edu/~peterp/707.html contains homework assignments, computational assignments, and anything else the instructor wants the students to know about. Please check this page often for new material.