18.2.1. Consistency

Try these first

1. 1.

   What is the local truncation error?

   Hint

   Think about what happens in a single step of approximation.

2. 2.

   How does the step size ℎ affect consistency?

   Hint

   Consider what happens as we make steps smaller.

3. 3.

   What does consistency in numerical methods refer to?

   • Local truncation error approaches zero as step size decreases
   • Numerical solutions being exact
   • Stability of numerical methods

   Hint

   Think about how error behaves as we reduce our step size.

4. 4.

   True or False: A method can be consistent and not convergent.

   • True
   • False

   Hint

   Reflect on the relationship between consistency and other properties.

5. 5.

   Given the function y' = -y, apply Euler's method with various step sizes (0.1, 0.01, 0.001) to compute the local truncation error at each step. Analyze how the error decreases.

   Hint

   Calculate at each step and compare the approximations to the actual solution.

6. 6.

   Devise a new numerical method and analyze its consistency and stability. Carry out a detailed error analysis comparing it to Euler's method.

   Hint

   Consider different functions and step sizes in your evaluations.