RK4 - Case study

• - Rk4 conditionally stable for oscillating spring
• • Not permanent oscillations: Slight decrease of magnitude through time
• - Large improvement of accuracy compared to implicit Euler

• Stability region: \(\displaystyle \left|1+h\lambda+\frac{h^2}{2}\lambda+\frac{h^3}{6}\lambda^3+\frac{1}{24}h^4\lambda^4\right| \leq 1\)

• In the case of oscillating spring: \(\lambda=i\sqrt{K/m}=i\omega\)
• Stable if \(\left|1+i h \omega-\frac{h^2}{2}\omega^2-i\frac{h^3}{6}\omega^3+\frac{h^4}{24}\omega^4\right| \leq 1\)
• • \(\Rightarrow 1-\frac{h^6\omega^6}{72}+\frac{h^8 \omega^8}{576} \leq 1\)
  • \(\Rightarrow h^6\,\omega^6\,(h^2\omega^2-8)/576 \leq 0\)

• • \(\Rightarrow h\leq\frac{2^{3/2}}{\omega}= \frac{2^{3/2}}{\sqrt{K/m}}\simeq \frac{2.8}{\sqrt{K/m}}\)