18.4. Stability of Common Methods

Try these first

1. 1.

   What is the stability function for Euler's Method?

   Hint

   Think about the general form of stability functions.

2. 2.

   Define A-stability.

   Hint

   What does it mean for a method to handle certain values of λ?

3. 3.

   What defines the condition of a method being A-stable?

   • It is stable for Re(λ) < 0
   • It requires large step sizes
   • It is stable for Re(λ) > 0

   Hint

   Think about the real part of λ and its relation to A-stability.

4. 4.

   True or False: All numerical methods are conditionally stable.

   • True
   • False

   Hint

   Consider what makes a numerical method reliable.

5. 5.

   Assess the stability of the Midpoint method if h = 1 and λ = -3. Show your calculations clearly.

   Hint

   Recall how to derive R(z) for the midpoint method.

6. 6.

   Discuss the impact of the Lax Equivalence Theorem on the design of numerical methods.

   Hint

   What does this mean for method selection when solving ODEs?