Transcription of Von Neumann Stability Analysis - MIT OpenCourseWare
1 Spring 2009 lecture 14 03/31/09. Von Neumann Stability Analysis Lax-equivalence theorem (linear PDE): Consistency and Stability convergence . (Taylor expansion) (property of numerical scheme). Idea in von Neumann Stability Analysis : Study growth of waves eikx . (Similar to Fourier methods). Ex.: Heat equation ut = D uxx Solution: 2t u(x, t) = e Dk e ikx =G(k) growth factor no growth if |G(k)| 1 k FD Scheme: Ujn+1 Ujn Ujn+1 2 Ujn + Ujn 1. =D . t ( x)2. (Explicit Euler) (Central). D t Ujn+1 = Ujn + r Ujn+1 2 Ujn + Uj 1. n . , r=. ( x)2. Insert u(x, tn ) = eikx into FD scheme: Ujn+1 = eik x j + r eik x (j+1) 2eik x j + eik x (j 1).. = (1 + r(eik x + e ik x 2))eik x j = G(k) eik x j Growth factor: G(k) = 1 2r (1 cos(k x)).
2 FD scheme stable, if |G(k)| 1 k Here: worst case: k x = G(k) = 1 4r Hence FD scheme conditionally stable: 1. r 2. (seen before). 1. Fast version: G 1 ei 2 + e i 2D. =D = (cos( ) 1). t ( x)2 ( x)2. G = 1 2r (1 cos ), = k x Ex.: Crank-Nicolson . Ujn+1 Ujn 1. n+1. Uj+1 2 Ujn+1 + Uj 1. n+1. Ujn+1 2 Ujn + Uj 1. n =D +. t 2 ( x)2 ( x)2. G 1 1 ei 2 + e i . = D (G + 1) . t 2 ( x)2. 1 r (1 cos ). G=. 1 + r (1 cos ). Always |G| 1 unconditionally stable. Ex.: 2D heat equation ut = uxx + uyy Forward Euler n+1. U1j Uijn Uin+1,j 2Ui,jn n + Ui 1,j n Ui,j+1 n 2Ui,j n + Ui,j 1. = +. t ( x)2 ( y)2. x u(x, y, tn ) = ei(k,l) ( y ) = eikx eily G 1 eik x 2 + e ik x eil y 2 + e il y = +.
3 T ( x)2 ( y)2. t t G=1 2 2. (1 cos(k x)) 2 (1 cos(l y)). ( x) ( y)2. t t Worst case: k x = = l y G = 1 4 2. 4. ( x) ( y)2. Stability condition: 1. h2.. t t 1 1 1 1. + t + =. ( x)2 ( y)2 2 2 ( x)2 ( y)2 4. if x=h= y 2. MIT OpenCourseWare Numerical Methods for Partial Differential Equations Spring 2009. For information about citing these materials or our Terms of Use, visit.
