18.5. Example Problems

Try these first

1. 1.

   What does stability in numerical methods refer to?

   Hint

   Think about what happens to a ship in rough waters.

2. 2.

   What is the significance of the Lax Equivalence Theorem?

   Hint

   Remember the relationship between stability and convergence.

3. 3.

   What is the stability condition for a numerical method?

   • |𝑅(ℎ𝜆)| ≤ 1
   • |𝑅(ℎ𝜆)| > 1
   • |𝑅(ℎ)| = 1

   Hint

   Remember, if the stability function is greater than 1, errors will grow.

4. 4.

   True or False: A stable method guarantees convergence.

   • True
   • False

   Hint

   Consider the Lax Equivalence Theorem.

5. 5.

   Derive the stability condition for a modified Euler method applied to the equation 𝑦' = -𝜆𝑦, where 𝜆 > 0.

   Hint

   Start by expressing the numerical iteration and finding the corresponding stability function.

6. 6.

   Apply the Backward Euler method to the equation 𝑦' = -2𝑦 and discuss its properties, leveraging A-stability.

   Hint

   Evaluate the backward difference and its relation to the stability function.