18.2.3. Convergence

Try these first

1. 1.

   What does convergence mean in numerical methods?

   Hint

   Think about what happens as you take smaller and smaller steps.

2. 2.

   What is the Lax Equivalence Theorem?

   Hint

   Remember how consistency and stability relate to one another.

3. 3.

   What does convergence ensure in numerical methods?

   • Errors grow exponentially
   • Solution approaches exact solution
   • Step size has no effect

   Hint

   Think about the relationship between accuracy and step size.

4. 4.

   True or False: Stability is not necessary for establishing convergence.

   • True
   • False

   Hint

   Reflect on the connection between stability and convergence.

5. 5.

   Given a numerical method that failed to converge due to stability issues, identify and analyze the parameters contributing to this instability.

   Hint

   Refer to the stability function of the method in question.

6. 6.

   Develop a numerical method that accomplishes convergence for a given initial value problem and validate it with a specified tolerance.

   Hint

   Adjust the step size dynamically to meet the tolerance level specified in the problem.