I am trying to find whether the following is stable absolutely using the improved Euler and the Adams-Bashforth 2 scheme,
$u'=\begin{bmatrix} -20&0&0\\ 20&-1&0\\0&1&0\end{bmatrix}u=Au$, where the timestep is $\frac{1}{2}$.
From my notes we know that for $u'=\lambda u$ is absolutely stable iff $|1+h\lambda|<1$. But I am confused as to what the `$\lambda$' is. Is it simply the eigenvalue(s) of the problem, which then confuses me if we are using other methods, or is it just a scalar? If it is the eigenvalues, then how to do you derive the absolute stability for the system using the Adams-Bashforth 2 scheme?
So far I found, for Euler, that $\lambda=-20,-1,-0.1$ which is not stable for our timestep. However for deriving the stability for the Adams-Bashforth I got as far as,
$$u^{n+1}=u^nA(1-\frac{A}{2}+\frac{A^2}{4})$$
but struggle as to how to proceed. Any advice would be appreciated.
Best Answer
In the improved Euler method, your ODE gives the step \begin{align} k_1&=Au_n\\ k_2&=A(u_n+hk_1)=(A+hA^2)u_n\\ u_{n+1}&=u_n+\frac h2 (k_1+k_2)=(I+hA+\frac{h^2}2A^2)u_n \end{align} What you want for stability is that the matrix factor is contracting if all eigenvalues of $A$ have negative part. That means you need to find $h$ so that $|1+(1+hλ)^2|<2$ for all relevant eigenvalues.
In the most simple Adams-Bashford method $u_{n+1}=u_n+\frac h2(3Au_n-Au_{n-1})$ you actually need to solve the linear recursion problem. Considering the eigenspaces this again reduces to an equation for the eigenvalues, the characteristic equation of the recursion for an eigenvalue $λ$ is $$ q^2=q+\frac h2(3λq-λ) $$ which has roots around $q=1+hλ$ and $q=\dfrac{hλ}{2+3hλ}$, compute the exact values with the solution formula, which both need to have absolute value smaller $1$.