How Do You Determine the Region of Absolute Stability for the Theta Method?

[eckiller]

Hello,

I am given the method: y_(n+1) = y_n + h f(t_n + w h, (1-w)y_n + w y_(n+1).

I am to determine the region of absolute stability;

I am also to determine for which w in [0, 1] is the method A(a) stable,
i.e., the region of absolute stability contains a sector about the negative
real axis.

I found the root of the characteristic polynomial in the complex plane to
be:

z = [1 + h k (1-w) ] / [1 - h k w]

So RAS = {hk : |z| < 1}

Can I simply that further? I.e., can I get a more explicity formula for
what the hk that satisfy z < 1 is?

Also I am at a loss on how to solve for when the method is A(a) stable.
Please help if you can. Thanks in advance.

 

[M98Ranger]

Hi there,

Thank you for sharing your work so far. The Theta method is a commonly used numerical method for solving differential equations. To determine the region of absolute stability, we need to find the values of hk that satisfy |z| < 1. This means that the absolute value of the root of the characteristic polynomial must be less than 1 for the method to be stable.

To simplify the expression for z, we can use the fact that the absolute value of a complex number z can be written as |z| = sqrt(z* z). Applying this to our expression for z, we get:

|z| = sqrt([1 + h k (1-w) ] / [1 - h k w] * [1 + h k (1-w) ] / [1 - h k w])

= sqrt([1 + h k (1-w)]^2 / [1 - h k w]^2)

= (1 + h k (1-w)) / (1 - h k w)

Now, for the method to be A(a) stable, the region of absolute stability must contain a sector about the negative real axis. This means that the values of hk must satisfy the inequality |z| < 1 for all values of k in the range [-a, 0]. This can be written as:

|z| < 1 for all k in [-a, 0]

(1 + h k (1-w)) / (1 - h k w) < 1 for all k in [-a, 0]

(1 + h k (1-w)) < (1 - h k w) for all k in [-a, 0]

Solving for h k, we get:

h k < [1 - (1 + h k (1-w))] / (1 - w)

= [1 - 1 - h k + h k w] / (1 - w)

= - h k w / (1 - w)

Therefore, the method is A(a) stable for all values of hk satisfying:

h k < - h k w / (1 - w) for all k in [-a, 0]

I hope this helps. Good luck with your further analysis!