Math 6610 • Analysis of Numerical Methods I

{Ordinary Differential Equations}
Consider the initial value problem for a system of two ODEs
\[
  y'=Ay
\]
where \(A\) has eigenvalues \(-10^4\) and \(1\). Which of the following schemes would be
the best choice for solving this problem? Justify your answer in terms of stability,
accuracy, and efficiency.
\[
  y^{n+1}=y^n+hAy^n, \tag{1}
\]
\[
  y^{n+1}=y^n+hAy^{n+1}, \tag{2}
\]
\[
  y^{n+1}=y^n+\frac{h}{2}(Ay^n+Ay^{n+1}). \tag{3}
\]