Stability analysis of Implicit Euler

• What are the general conditions for which \(u'(t)=\lambda\,u(t)\) converge using Implicit Euler ?

• \(u^{k+1}=u^{k}+h\,\lambda\,u^{k+1}\)
• \(\Rightarrow (1-h\,\lambda)\,u^{k+1}=u^{k}\)
• \(\Rightarrow u^{k+1}=\frac{1}{1-h\,\lambda}\,u^{k}\)

• Stability condition: \(\displaystyle \left|\frac{1}{1-h\lambda}\right|\leq 1 \Rightarrow \left|1-h\lambda\right|\geq 1\)

• \(|1/h-\lambda|\geq 1/h\): exterior of a disc centered on \((1/h,0)\) with radius \(1/h\)

• \(\Rightarrow\) Fully enclose \(\mathbb{R}_e(\lambda)\leq 0\)
• \(\Rightarrow\) Implicit euler is unconditionally stable

• Rem. May converges even when the true solution does not.