AWR8PbTJloe-n5dUiTHGFAdYIPXXKLd_Fyt8khGflhQ66CMuAaY3_BBCdrbZwYO6EfxcmuLLXF2wzI8iXjH6BA

Lecture Note Sketches




3 views

Unformatted text preview:

Lecture Note Sketches Spectral Methods for Partial Differential Equations Hermann Riecke Engineering Sciences and Applied Mathematics [email protected] June 3, 2009 1 Contents 1 Motivation and Introduction 8 1.1 Review of Linear Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Approximation of Functions by Fourier Series 12 2.1 Convergence of Spectral Projection . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.2 The Gibbs Phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.3 Discrete Fourier Transformation . . . . . . . . . . . . . . . . . . . . . . . . . . 25 2.3.1 Aliasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 2.3.2 Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 3 Fourier Methods for PDE: Continuous Time 34 3.1 Pseudo-spectral Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.2 Galerkin Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4 Temporal Discretization 38 4.1 Review of Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.2 Adams-Bashforth Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.3 Adams-Moulton-Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 4.4 Semi-Implicit Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.5 Runge-Kutta Methods . . . . . . . . . . . . . . . . . . . . . . ...





Loading Unlocking...

Login

Join to view Lecture Note Sketches and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?

Sign Up

Join to view Lecture Note Sketches and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?