Math 6610 • Analysis of Numerical Methods I

{Numerical ODEs}

\textbf{(a)} Consider the general system of ODEs for \(y(x)\in\R^n\), \(x\in\R\) and
\(f=[f_1,\ldots,f_n]^T\):
\[
  y'=f(x,y),
\]
where we assume \(f\) to be Lipschitz continuous. To study the numerical stability, we
linearize \(f\) near \(x=x_0,y=y_0\) to obtain:
\[
  y'\approx f(x_0,y_0)+f_x(x_0,y_0)(x-x_0)+\nabla_y f(x_0,y_0)(y-y_0).
\]
If we denote \(A=\nabla_y f(x_0,y_0)\), then we claim that the local stability of the
above system can be analyzed by studying the following constant coefficient system
\[
  y'=Ay.
\]
Justify this claim by analyzing the simple scalar equation
\[
  y'=a+bx+cy,
\]
where \(a,b,c\) are constants, in the explicit Euler's method, to show that the
stability is determined by how we solve \(y'=cy\). 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)} 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.