Lec15 - Fourier and Chebyshev Spectral Methods for BV...

Info iconThis preview shows pages 1–6. Sign up to view the full content.

View Full Document Right Arrow Icon
Fourier and Chebyshev Spectral Methods for BV Problems: Fast Fourier Transforms Think globally. Act locally -- L. N. Trefethen, “Spectral Methods in Matlab” (SIAM, 2000)
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Lecture 15 2 Global Interpolation z Remember that when we studied the subjects of: Numerical differentiation Numerical integration z We found that much higher accuracy could be obtained for smooth functions if the approximants were based on a polynomial global interpolation using all the data across an interval z This was called “spectral accuracy” z So far, we have only used local differentiation formulas to solve BV problems (finite differences) with algebraic O(h 2 ) errors z In this lecture we will use global or “spectral” methods to solve BV problems with much higher accuracy z The global interpolation will be based on either: Chebyshev polynomial interpolation Trigonometric Fourier interpolation c i – coefficients, e i (x) basis functions
Background image of page 2
Lecture 15 3 Chebyshev polynomial interpolation z Given N+1 data points (x i ,y i ), there is a unique polynomial of degree N that goes through all the points z In this lecture you will learn why the Chebyshev form of the polynomial is the most useful: Chebyshev polynomial expansion Chebyshev recursion formula z We obtain the expansion coefficients {c} by “collocation” at the N+1 Chebyshev points: Chebyshev points
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Lecture 15 4 Errors and Intervals z A remarkable feature of global interpolation with Chebyshev polynomials is the rapid rate of convergence. If the function being described is suitably smooth (analytic), the error in the approximant can be shown to decay as fast as z This exponentially fast convergence with the number of data points is referred to as “spectral accuracy” and is highly desirable z To achieve these results, it is very important that the independent (x) variable of your data be scaled to the interval [-1,1] or Chebyshev interpolation will not work! z A simple change of variable will do the trick:
Background image of page 4
Lecture 15 5 Chebyshev Differentiation z Recall how we do Chebyshev differentiation of a function y(x) sampled on the Chebyshev grid, i.e. { y}=(y 0 , … , y N ) T : Let p(x) be the unique polynomial of degree N with p(x j ) =y j , 0 j N Set y (1) j = p 0 (x j ) [derivative dy/dx at grid points] z This is a linear operation with an (N+1)x(N+1) matrix [D N ]: z The second derivative on the Chebyshev points is simply:
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
Image of page 6
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 12/29/2011 for the course CHE 132b taught by Professor Ceweb during the Fall '09 term at UCSB.

Page1 / 22

Lec15 - Fourier and Chebyshev Spectral Methods for BV...

This preview shows document pages 1 - 6. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online