Von Neumann Stability Analysis - MIT OpenCourseWare

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)).

18.336 spring 2009 lecture 14 03/31/09 Von Neumann Stability Analysis Lax-equivalence theorem (linear PDE): Consistency and stability ⇐⇒ convergence

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