Spring 2018 • Question 5
**Absolute stability for ODEs**
For the ODE \(y'=f(y)\), and given \(p>0\), a linear multistep scheme has the form
\[
\sum_{j=0}^{p}a_jy_{n+j}
+
\Delta t\sum_{j=0}^{p}b_jf(y_{n+j})=0
\]
where \(\Delta t\) is the timestep, \(a_p=1\), and \(y_n\) is the numerical
approximation to \(y(t=n\Delta t)\), where the starting time is \(t=0\).
**a.** Define the region of absolute stability for the above scheme.
**b.** Compute and sketch the region of absolute stability for the backward
(i.e., implicit) Euler scheme.