18.2.2. Stability

Try these first

1. 1.

   What is stability in numerical methods?

   Hint

   Think about how errors affect the numerical solution.

2. 2.

   Define absolute stability.

   Hint

   Consider the meaning of a bounded growth of errors.

3. 3.

   What determines the stability of a numerical method?

   • A) The method's accuracy
   • B) The behavior of errors under perturbations
   • C) The initial conditions

   Hint

   Think about what happens when perturbations occur.

4. 4.

   True or False: A stable method will always provide the correct solution.

   • True
   • False

   Hint

   Remember the relation between stability and convergence.

5. 5.

   Consider the differential equation $y' = -2y$ solved by Euler's method. If you choose a step size of $h = 0.5$, is this method stable? Justify your answer using the stability function.

   Hint

   Calculate the stability function and check the modulus.

6. 6.

   Given a multistep method, analyze its stability region and identify if it would be suitable for stiff ODEs. Explain your reasoning.

   Hint

   Determine the relationship between roots of the characteristic equation and the unit circle.