A Short Introduction to Pseudo-Spectral Methods - Part 1

DATES: 09 - 13 November 2020 (5 sessions)
TIME: 09:00 - 11:00 (a total of 10 hours)
LOCATION: Online

ABSTRACT:
An important class of very accurate numerical methods for solving differential equations is the pseudo-spectral methods. One of the main reasons is because it involves the approximation of solutions through trial (basis) functions which are evaluated on the whole domain rather than only a part of it. The applications of these techniques are ubiquitous in fluid dynamics, non-linear waves, seismic modeling, etc.

In this course, basics of the pseudo-spectral collocation methods will be covered. We will restrict to the case when the trial functions are trigonometric and Chebyshev polynomials. Apart from the relevant theory, each lecture will be supplemented with coding examples and exercises implemented in Octave. Finally, we will survey some recent applications of these methods.

The course will be mainly aimed at masters and Ph.D. students but anyone with a basic knowledge of ODEs/PDEs, linear algebra, Fourier transform, complex and numerical analysis should be able to follow it.

PROGRAMME:
The following topics are intended to be covered:
- Motivation and Introduction
- Differentiation matrices, unbounded grids, Fourier transforms.
- Periodic and non-periodic grids.
- Application to some boundary values problems


REFERENCES:
[1] Lloyd N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2000.
[2] C. Canuto, M. Y. Hussaini, A. Quarteroni, T. A. Zang, Spectral Methods in Fluid Dynamics, Springer, Berlin, Heidelberg, 1988.
[3] F. de la Hoz, S. Kumar, L. Vega, On the Evolution of the Vortex Filament Equation for regular M-polygons with nonzero torsion, SIAM Journal on Applied Mathematics, 80(2), pp. 1034-1056, 2020.


*Registration is free, but mandatory before November 6th.
To sign-up go to https://forms.gle/8oBu9kuEs3kmkdDo9 and fill the registration form.

• BCAM-UPV/EHU_Course_2020_11_09