Math 6610 • Analysis of Numerical Methods I

{ODEs}

Consider the initial value problem for a system of ODEs
\[
  y'=Ay
\]
where \(A\) has eigenvalues with real parts ranging from \(-10^6\) to \(-1\). Which of
the following schemes would be the best choice for solving this problem? Justify
your answer in terms of stability, accuracy, and efficiency. Does your answer
change depending on the size and condition number of \(A\)? Explain your answer.
\[
  y^{n+1}=y^n+kAy^n, \tag{1}
\]
\[
  y^{n+1}=y^n+kAy^{n+1}, \tag{2}
\]
\[
  y^{n+1}=y^n+\frac{k}{2}(Ay^n+Ay^{n+1}). \tag{3}
\]

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