18.2. Consistency, Stability, and Convergence

Try these first

1. 1.

   Define consistency in the context of numerical methods.

   Hint

   Think about how the accuracy of numerical methods improves with smaller steps.

2. 2.

   What does a stable numerical method imply?

   Hint

   Consider what we do with rounding errors.

3. 3.

   What is the definition of consistency in numerical methods?

   • The method remains accurate regardless of step size
   • The local truncation error decreases as step size decreases
   • Stability is guaranteed

   Hint

   Look for the relationship between step size and accuracy.

4. 4.

   Is a stable method guaranteed to converge?

   • True
   • False

   Hint

   Recall the Lax Equivalence Theorem.

5. 5.

   Consider a numerical method that has a consistent local truncation error but demonstrates growing errors due to rounding each step. Discuss the potential challenges faced in applying this method.

   Hint

   Focus on the implications of error growth in larger calculations.

6. 6.

   Demonstrate the implications of the Lax Equivalence Theorem in practical numerical methods by providing an example of a method that fails to converge due to lack of stability.

   Hint

   Reflect on common methods in numerical algorithms and identify characteristics.