Stability analysis of explicit Euler
- \(u'(t)=\lambda\,u(t)\)
- \(\Rightarrow u^{k+1}=u^k+\lambda\,u^k\) using explicit Euler
- \(\Rightarrow u^{k+1}=(1+\lambda\,h)\,u^k\)
- \(\Rightarrow\) Stable if \(|1+\lambda\,h|\leq 1\) (conditionnal stability)
- \(|1/h+\lambda|\leq 1/h\): Interior of a disc centered on \((-1/h,0)\) with radius \(1/h\)
- Rem. For 1D elastic spring
- \(\lambda = \pm i\sqrt{K/m}\)
- \(\Rightarrow\) \(|1+i\sqrt{K/m} h|=\sqrt{1+K/m\,h^2}>1\)
- \(\Rightarrow\) Explicit euler is always unstable on the elastic spring problem.
