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Mathematics Qualifying Exam Question Bank

Department of Mathematics

Question

Math 6620 • Analysis of Numerical Methods II

Summer 2024 • Question 4

Question

Math 6620 • Analysis of Numerical Methods II Summer 2024 • Question 4

(20 pts)

  1. (10 pts) Use von Neumann stability analysis to determine a stability condition for, \[ D^+u_j^n = D_0u_j^n, \] where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\), \(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and \(D_\pm u_j^n = \pm (u_{j\pm 1}^n-u_j^n)/h\).
  2. (10 pts) Use von Neumann stability analysis to determine a stability condition for \[ D^+u_j^n = D_+D_-u_j^n. \]

Raw LaTeX

(20 pts)
\begin{enumerate}

\item
(10 pts) Use von Neumann stability analysis to determine a stability condition for,
\[
D^+u_j^n = D_0u_j^n,
\]
where \(D^+u_j^n = (u_j^{n+1}-u_j^n)/k\),
\(D_0u_j^n = (u_{j+1}^n-u_{j-1}^n)/(2h)\), and
\(D_\pm u_j^n = \pm(u_{j\pm1}^n-u_j^n)/h\).



\item
(10 pts) Use von Neumann stability analysis to determine a stability condition for
\[
D^+u_j^n = D_+D_-u_j^n.
\]


\end{enumerate}

Topic Tags

Absolute Stability and A-Stability Consistency, Stability, and Convergence Von Neumann Stability Analysis Zero-Stability and Root Conditions