Math 6610 • Analysis of Numerical Methods I

{Numerical Methods for ODEs}
Consider the Linear Multistep Method
\[
  y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}hf_{n+2}
\]
for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta\). You may assume that
\(f\) is Lipschitz continuous with respect to \(y\) for all \(x\). \textbf{a)} Analyze the
consistency, stability, accuracy, and convergence properties of this method.
\textbf{b)} Would this method be a reasonable one to use for the initial value
problem:
\[
  y'=-10^8[y-\cos(x)]-\sin(x),\qquad y(0)=1?
\]
Justify your answer.