Numerical methods for systems of differential equations

[mkerikss]

Homework Statement

Consider the implicit (not actually sure wether that's the correct english word, my material is in Finnish and I'm Swedish-speaking ) method
x_j+1=x_j+h/2(f(tj,xj)+f(t_j+h,x_j+1))

a)Write an appropriate Runge-Kutta scheme
b) What is the methods rank when we use the equation x'=ax ? I have absolutely no clue if rank is the correct word but it is a number that describes some kind of order. Unfortunately I can't describe it better than that because I don't really know what it is and that is one of my questions.
c)What is the methods stability area?

The Attempt at a Solution

In a), I'm not sure if the different numerical methods have different schemes, but one I've used in an earlier exercise is

__t____________t₁_____t₂_______t_n
x₁
x₂
:
:
x_n

Is this correct?

In b), I'm really lost. I've figured out that x'=f(x)=ax. The one previous time I've done an exercise about "rank" the solution was to use Taylor expansions, and then all x, x', x'' eventually disappeared, which if I recall correctly implies that the rank is 2. But I have no idea how to do that in this particular exercise.

I've also seen one example of calculating stability areas. Just to check that I have understood correctly, does this mean that I can write an equation where x_j+1=Aⁿx₀, solve A:s eigenvalues and their absolute values have to be <1. Then I can solve h. Or is this a different situation?

Thanks for your help!

 

[mkerikss]

Anyone? I'm preparing for a test tomorrow and would appreciate your help