AM256 Numerical Solution of Partial Differential Equations II
Spectral Methods for Time-Dependent Problems
Instructor: Professor Jan S. Hesthaven
Computer Teaching Assistant: TBD
Where and When: I-hour (tue/thu, 10:30-11:50).
Barus and Holley room 160
Office Hours: TBD
Textbook: J. S. Hesthaven, S. Gottlieb, and D. Gottlieb, 2007,
Spectral Methods for Time-Dependent Problems. Cambridge Monographs
on Applied and Computational Mathematics 21,
Cambridge University Press, Cambrigde, UK
Description:
During the last decades, spectral
methods have been developed and applied for solving partial differential
equations from most areas of physics and engineering, often with remarkable
success, yielding vastly better results than what is obtainable with more
traditional methods, e.g., finite difference or finite element schemes.
In this course we shall develop the basic theory underlying the formulation
of spectral methods suitable for the solution of time-dependent
partial differential equations as they appear in science and engineering.
Emphasis shall be on the theoretical foundation as well as on
computational aspects.
Syllabus:
- Introduction: Motivation for the use of high-order
methods. Phase error analysis. Convergence, consistency and stability.
Method of weighted residuals. Elements of necessary theoretical
background and functional theory.
- Fourier Methods for Smooth Problems: Fourier series.
Spectral and pseudospectral Fourier methods. Basic approximation results
and stability of Fourier methods.
- Polynomial Methods for Smooth Problems: Polynomial expansions
and Jacobi polynomials. Gauss quadrature rules.
Spectral and pseudospectral polynomial methods. Boundary conditions.
Approximation and stability theory.
- Issues for Nonsmooth Problems: Gibbs phenomenon,
filtering and stabilization. Implementation issues.
- Numerical/Implementation Aspects: Fast Fourier Transforms.
Even-Odd decompositions. Effect of round-off errors. Use of mappings.
- Temporal Integration Schemes: Discrete stability and
eigenvalues. Explicit and implicit time-stepping schemes.
Prerequisites: Programming skills in C/Fortran/Matlab is
assumed as is basic knowledge about partial differential equations and
finite difference solution of such problems
corresponding to AM255.
Homework: There will generally be weekly homework
assigned each Thursday and due the following Thursday.
Late homework will be accepted only under special circumstances and
with prior approval and will then only count at most 50 % of
its full share towards the final grading. The homework will include
theoretical as well as computational exercises, i.e., some computer
programming is expected as part of most homework.
Exams: There will be one midterm and
one take-home final.
Grading: The grading will be based on a weighting
of 50 % on the homework, 20 % on the midterm, and 30 % on the
final.
