Day04-slides - Stability Atms 502 CS 505 CSE 566 Numerical...

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1 Atms 502 CS 505 CSE 566 Numerical Fluid Dynamics Tue., 5 September 2006 9/29/06 Atms 502 - Fall 2006 - Jewett 2 Stability Consider any solution as Fourier series Examine behavior of one component if every Fourier component is stable … i.e. every possible wave’s amplitude is bounded then our scheme must be stable . Series: u j n = a k n e ikj " x k = # N N $ % u j n = e ikj " x ; u j n + 1 = A k e ikj " x General series Initial condition for wavenumber “k” Solution After one time step 9/29/06 Atms 502 - Fall 2006 - Jewett 3 Back up: review About that initial state … This has amplitude 1 and phase θ .. So we’re assuming u is some kind of sinusoidal function - here, one wave component, with wavenumber k. Wave length L = 2 π /k so: kx has units of an angle. u j n = e ikx ; x " j # x ; u j n = e ik j # x ( ) e i " = cos " + i sin " kx = 2 " x L 9/29/06 Atms 502 - Fall 2006 - Jewett 4 Amplification factor After one step the solution will be: A k is amplification factor for wavenumber k . Really the amplitude is not 1, but a k We can relate the amplitude of wave number k to amplitudes at earlier times, back to t=t 0 . u j n + 1 = A k e ikj " x a k n = A k a k n " 1 = ... = A k [ ] n a k 0
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2 9/29/06 Atms 502 - Fall 2006 - Jewett 5 Stability criteria |A k | 1: what does this mean? The numerical solution results, over time, remain bounded by their initial values.
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