Math 6610 • Analysis of Numerical Methods I

{Numerical ODEs}

\textbf{a)} (8\%) Consider the general system of ODEs for
\(y(x)\in\mathbb{R}^n\), \(x\in\mathbb{R}\), and
\[
  f=[f_1(x,y),\ldots,f_n(x,y)]^T:
  \qquad
  y'=f(x,y),
\]
where we assume \(f\) is continuously differentiable in both \(x\) and \(y\). To study
local stability, we compare two solutions \(y_1(x)\) and \(y_2(x)\) where
\(y_1(x_0)\approx y_2(x_0)\). Derive the linearized system for
\(e(x)=y_1(x)-y_2(x)\) near \(x=x_0\) to show that
\[
  e'\approx Ae
\]
and find this square matrix \(A\). What is the most important property of \(A\) we need
to find that will affect the choice of numerical schemes and step sizes?

\textbf{b)} (12\%) Now consider the initial value problem
\[
  y''+2\mu y'+(1+\mu^2)y=0,
\]
\[
  y(0)=0,\qquad y'(0)=1,
\]
where \(\mu>0\) is a given constant. If we use a simple explicit Euler's method,
determine the maximum step size \(h\) that is allowed for the solution to be stable.