18.6. Summary

Try these first

1. 1.

   What ensures that the local error of a numerical method approaches zero as the step size decreases?

   Hint

   Think about the error shrinking property.

2. 2.

   What type of stability applies to methods like Backward Euler?

   Hint

   Recall the highest level of stability definition.

3. 3.

   What does consistency in a numerical method refer to?

   • It ensures local error approaches zero.
   • It guarantees results are accurate.
   • It relates to stability only.

   Hint

   Think about the shrinking error concept.

4. 4.

   True or False: A-stability guarantees that numerical methods are stable for Re(λ) < 0.

   • True
   • False

   Hint

   Recall the definition of A-stability.

5. 5.

   Consider a numerical method that is consistent but unstable. Describe what issues may arise when solving a stiff ODE with this method, and propose a solution to avoid those issues.

   Hint

   Think about the nature of stiffness in ODEs and how certain methods cope.

6. 6.

   Analyze the difference in stability regions for Euler's Method and Backward Euler. Use a graphical representation to illustrate your point.

   Hint

   Graphically represent complex plane regions that define where R(z) holds.