17.1.5. Stability and Convergence

Try these first

1. 1.

   Define stability in the context of numerical methods.

   Hint

   Think about how errors affect the results over time.

2. 2.

   What does convergence mean?

   Hint

   Consider what happens when you make the step size smaller.

3. 3.

   What does stability in numerical methods ensure?

   • Errors grow indefinitely
   • Errors are controlled
   • No errors occur

   Hint

   Remember the definition of stability.

4. 4.

   True or False: Convergence means the numerical solution moves further away from the exact solution as the step size decreases.

   • True
   • False

   Hint

   Think about the goal of solving ODEs.

5. 5.

   Consider a numerical method where a large truncation error occurs at each step. Explain how this impacts stability and convergence.

   Hint

   Think about how errors compound in numerical methods.

6. 6.

   You are tasked with solving a stiff ODE. What steps can you take to ensure your method is both stable and convergent?

   Hint

   Consider the properties of methods suitable for stiff problems.