Math 6610 • Analysis of Numerical Methods I

{Numerical Methods for ODEs:}

Consider the Linear Multistep Method
\[
  y_{n+2}-\frac{4}{3}y_{n+1}+\frac{1}{3}y_n=\frac{2}{3}k f_{n+2}
\]
for solving an initial value problem \(y'=f(y,x)\), \(y(0)=\eta\). You may assume
that \(f\) is Lipschitz continuous with respect to \(y\) uniformly for all \(x\).

\partitem{a} Analyze the consistency, stability, accuracy, and convergence
properties of this method.

\partitem{b} Sketch a graph of the solution to the following initial value
problem.
\[
  y'=-10^8[y-\cos(x)]-\sin(x),\qquad y(0)=2.
\]
Would it be more reasonable to use this method or the forward Euler method for
this problem? What would you consider in choosing a timestep \(k\) for each of the
methods? Justify your answer.

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