Numerical DE: 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.

 

[gtoguy97]

To determine the region of absolute stability, you need to find the roots of the characteristic polynomial. This is done by solving for the value of z in:z = [1 + h k (1-w) ] / [1 - h k w].The region of absolute stability is then defined as the set of hk values for which the magnitude of the root is less than 1. That is, RAS = {hk : |z| < 1}.To determine for which w in [0, 1] the method is A(a) stable, you'll need to analyze the stability of the method for different values of w. This can be done by examining the behavior of the roots of the characteristic polynomial as w varies. If the roots stay within the unit circle for all values of w in the range [0, 1], then the method is A(a) stable.

 

[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!