This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 $...
View
Full
Document
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
 JEWETT

Click to edit the document details