1 / 32

Lecture 22

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.

tao
Télécharger la présentation

Lecture 22

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Lecture 22 MA471 Fall 2003

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

  3. 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

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

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

  6. Code for the DFT

  7. Code for Inverse DFT

  8. Fast Fourier Transform • See handout

  9. Spectral Derivative • We can differentiate the interpolant by:

  10. 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.

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

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

  13. Spectral Differentiation Scheme • Use an fft to compute: • Differentiate in spectral space. i.e. compute:

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

  15. 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:

  16. Spectral Differentiation Code Typo: See corrected Code on webpage • DFT data • Scale Fourier coefficients • IFT scaled coefficients

  17. 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:

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

  19. 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.

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

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

  22. Factorized Scheme

  23. JST + Advection Equation • We now use the PDE definition +

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

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

  26. 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

  27. FFT Resources • Check out: http://www.fftw.org/ for the fastest Fourier transform in the West • Great list of links: • http://www.fftw.org/links.html • FFT for irregularly spaced data: • http://www.math.mu-luebeck.de/potts/nfft/

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

  29. Upshot: 2D Advection/DFTUsing MPI_Alltoall Note: using fft the compute time will drop by approx. A factor of 30 (for N=256)

  30. Upshot: 2D Advection/DFTUsing Non-Blocked Sends Note: I isend and irecv, then compute the x-derivatives, then waitall, then compute the y-derivatives, then isend/recv and waitall (not the best option perhaps).

  31. With FFT • Replacing the DFT calls with FFT calls will radically reduce the computation time. • It will be much harder to hide the communication time behind the compute time. • Good luck 

More Related