Problems

1. Show that the operator associated with the difference method

   

   is of order 4 and its error constant is .

2. Determine the constants and in such a way that the operator associated with

   

   is of order 4, and determine the error constant. Verify that the resulting method is unstable.

3. Find the most accurate implicit linear two-step method. Find its principle part of the local truncation error.

4. Show that the order of the linear multistep method

   

   is zero. By finding the exact solution of the difference equation which arise when this method is applied to the initial value problem

   

   and demonstrate that the method is indeed divergent.

5. Show that the order of the linear multistep method

   

   is 2 if and is 3 if . Show that the method is zero-unstable if . Illustrate the resulting divergence of the method with by applying it to the initial value problem

   

   and solving exactly the resulting difference equation when the starting values are .

6. Find the range of for which the linear multistep method

   

   is zero-stable. Show that there exists a value of for which the method has order 4 but that if the method is to be zero-stable, its order cannot exceed 2.

7. If , find a of degree four such that the method has maximum order. What is that order and what is the error constant?

8. If . Find such that is of second degree and the order is two.

9. Find the region of absolute stability of the method given by

   

10. Find the interval of absolute stability for the two-step Adams-Bashforth method given by