Problems

1. Find the stiffness ratio for the system

   

   

   What is the largest step length which can be used with a fourth order Runge-Kutta method?

2. Show that the Trapezoidal rule is A-stable and the backward Euler method is L-stable.
3. Consider a one-parameter family of one-step methods given by

   

   Find its order and investigate A-stability of the method.

4. Consider a two-parameter family of two step methods given by

   

   Find its order and investigate A-stability of the method.

5. Show that the Trapezoidal rule is exponentially fitted at , and the backward Euler method is exponentially fitted at ..

6. Consider the backward differentiation methods

   

   for . Shows that these methods are A-stable.

7. Show that the following semi-explicit method of order two given by

   

   with

   

   

   is A-stable.