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Unformatted text preview: 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 AmpliFcation 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 . u j n + 1 = A k e ikj " x a k n = A k a k n " 1 = ... = A k [ ] n a k 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. Appropriate if the norm of the true solution is constant with time. u n " u 9/29/06 Atms 502 - Fall 2006 - Jewett 6 Measures of magnitude How we measure amplitude behavior : norms • For vector x- a chain of numbers - L 1 norm: sum of absolute values of all numbers L 2 norm: square root of sum of squares Infinite norm: maximum value: • There are similar relations for matrix norms x 2 " x j 2 # x " # max x i for 1 $...
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This note was uploaded on 10/06/2009 for the course ATMOSPHERI 502 taught by Professor Jewett during the Spring '09 term at University of Illinois at Urbana–Champaign.
- Spring '09