Lecture 22 MA471 Fall 2003

Advection Equation Recall the 2D advection equation: We will use a Runge-Kutta time integrator and spectral representation in space.

Periodic Data Let’s assume we are given N values of a function f at N data points on the unit interval. For instance N=8 on a unit interval: 0 1/8 2/8 3/8 4/8 5/8 6/8 7/8 8/8

Discrete Fourier Transform We can seek a trigonometric interpolation of a function f as a linear combination of N (even) trigonometric functions: Such that:

Transform The interpolation formula defines a linear system for the unknown fhat coefficients: Or:

Code for the DFT

Code for Inverse DFT

Fast Fourier Transform See handout

Spectral Derivative We can differentiate the interpolant by:

Detail We note that the derivative of the k=(N/2) mode is technically: However, as we show on the next slide – this mode has turning points at all the data points.

Real Component of N/2 Mode i.e. the slope of the k=(N/2) mode is zero at all the 8 points..

Modified Derivative Formula So we can create an alternative symmetric derivative formula:

Spectral Differentiation Scheme 1)Use an fft to compute: 2)Differentiate in spectral space. i.e. compute:

cont 3) Then: 4) Summary: a) fft transform data to compute coefficients b) scale Fourier coefficients c) inverse fft (ifft) scaled coefficients

Final Twist Matlab stores the coefficients from the fast Fourier transform in a slightly odd order: The derivative matrix will now be a matrix with diagonal entries:

Spectral Differentiation Code 1)DFT data 2)Scale Fourier coefficients 3)IFT scaled coefficients

Two-Dimensional Fourier Transform We can now construct a Fourier expansion in two variables for a function of two variables.. The 2D inverse discrete transform and discrete transform are:

Advection Equation Recall the 2D advection equation: We will use a Runge-Kutta time integrator and spectral representation in space.

Runge-Kutta Time Integrator We will now discuss a particularly simple Runge-Kutta time integrator introduced by Jameson-Schmidt-Turkel The idea is each time step is divided into s substeps, which taken together approximate the update to s’th order.

Side note: Jameson-Schmidt- Turkel Runge-Kutta Integrator Taylor’s theorem tell’s us that We will compute an approximate update as:

JST Runge-Kutta The numerical scheme will look like We then factorize the polynomial derivative term:

Factorized Scheme

JST + Advection Equation We now use the PDE definition +

Now Use Spectral Representation A time step now consists of s substages. Each stage involves: a)Fourier transforming the ctilde using a fast Fourier transform (fft) b)Scaling the coefficients to differentiate in Fourier space c)Transforming the derivatives back to physical values at the nodes by inverse fast Fourier transform (ifft). d)Finally updating ctilde according to the advection equation. At the end we update the concentration.

Matlab Implementation First set up the nodes and the differentiation scalings.

24) Compute dt 26) Initiate concentration 28-29) compute advection vector 32-45) full Runge- Kutta time step 35) fft ctilde 37) x- and y- differentiation in Fourier space 40-41) transform back to physical values of derivatives 43) Update ctilde

FFT Resources Check out: for the fastest Fourier transform in the Westhttp:// Great list of links: – FFT for irregularly spaced data: –

Lab Exercise 1)Download example DFT code from website. 2)Benchmark codes as a function of N 3)Write your own version using an efficient FFT (say fftw) which: 1) FFT’s data 2)Scales coefficients 3)IFFT’s scaled coeffients 4)Write an advection code using the JST Runge- Kutta time integrator. This is the scalar version of Project 4