Rawitscher Minicourse on Numerical Spectral Methods
Title: Numerical Spectral Methods for Solving Differential or Integral Equations
Lecturer: George Rawitscher (Univ. of Connecticut)
Dates: March 18 – April 15, 2015
Place: Third floor of IFT-UNESP – Room 2
Times: Mondays, Wednesdays and Fridays at 10 am.
The main purpose of this course is to inform the scientific community of the spectral method for computing algorithms. Spectral methods have been introduced in the 70´s, but their power is not yet fully appreciated by the scientific community. The main difference from previous methods, such as finite difference and finite elements, is that the value of a function at all the mesh-points is taken into account simultaneously, while for the finite difference methods only three or five points are included at one time.
After some initial reminders about computational accuracy, the general properties of spectral methods will be described, based on the book by Lloyd. N. Trefethen “Spectral Methods in MATLAB” (SIAM, 2000). This will take six or seven lectures. After that, in the remaining six or five lectures, various applications of the spectral method will be presented, with special emphasis to the solution of the one-dimensional Schroedinger equation. Homework problems will be provided, but handing in the solutions is entirely optional.
The course will consist of a series of 12 lectures, taught three times per week. Its main purpose is to familiarize the audience with spectral methods that are now supplanting the more conventional and less efficient numerical methods. The course is directed mainly at physicists, but it would also be of interest to chemists and biologists.
There will be no application form for this activity and everyone is welcome to participate. For more information, send email to firstname.lastname@example.org.
Lecture 1. Numerical errors, Round-off and algorithm errors, Examples
Lecture 2. Finite difference methods, Euler, Numerov, Runge-Kutta
Lecture 3. Galerkin and Collocation methods. Examples
Lecture 4. a) Coefficient method; b) Differentiation matrices
Lecture 5. Mesh-points: Equispaced versus nonequi-spaced. Interpolation, integration
Lecture 6. Chebyshev Polynomials
Lecture 7. Convergence rate of expansions
Lecture 8. Spectral methods. Based on the book by Lloyd N. Trefethen “Spectral Methods in MATLAB”
Lecture 9. Spectral Finite Element Method
Lecture 10. Phase Amplitude representation of a wave function
Lecture 11. Three-body Equation ?
Lecture 12. Discussion