20.3. Stability and Convergence

Try these first

1. 1.

   Define stability in the context of numerical methods.

   Hint

   Think about how errors behave in the solution.

2. 2.

   What is the CFL condition?

   Hint

   Consider it as a limit on time steps.

3. 3.

   What is the primary purpose of establishing stability in numerical methods?

   • To ensure accuracy
   • To control errors
   • To simplify calculations

   Hint

   Consider what stability actually governs in terms of numerical output.

4. 4.

   True or False: Consistency guarantees that a numerical method will converge.

   • True
   • False

   Hint

   Think of the relationship between multiple conditions.

5. 5.

   Develop a numerical method to solve a simple PDE and discuss its stability, consistency, and convergence.

   Hint

   Use the CFL condition as a major point of reference.

6. 6.

   Critique an existing numerical method, identifying potential failures in stability or consistency and suggest improvements.

   Hint

   Conduct a thorough literature review on known methods.