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

• eckiller
In summary, to determine the region of absolute stability for the given method, you need to find the roots of the characteristic polynomial and solve for the hk values that satisfy |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 by examining the behavior of the roots of the characteristic polynomial.
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.

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.

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!

1. What is the Theta method in numerical differential equations?

The Theta method is a numerical technique used to solve initial value problems in differential equations. It is an extension of the popular Euler method and allows for a more accurate approximation of the solution by introducing a parameter, theta, which can be adjusted to control the accuracy and stability of the method.

2. How does the Theta method differ from other numerical methods?

The Theta method is a compromise between the explicit Euler method and the implicit trapezoidal method. It uses a combination of forward and backward Euler steps to approximate the solution, making it more accurate than the explicit method but less computationally expensive than the implicit method.

3. What is the stability of the Theta method?

The stability of the Theta method depends on the value of the parameter theta. For values of theta between 0 and 1/2, the method is conditionally stable, meaning that the step size must be small enough for the solution to remain stable. For values of theta between 1/2 and 1, the method is unconditionally stable, meaning that the step size can be larger without affecting the stability of the solution.

4. How do I choose the value of theta in the Theta method?

The value of theta can be chosen based on the desired accuracy and stability of the solution. A larger value of theta (closer to 1) will result in a more stable solution but may sacrifice some accuracy. A smaller value of theta (closer to 0) will result in a more accurate solution but may be less stable.

5. Can the Theta method be used for any type of differential equation?

Yes, the Theta method can be used for any type of differential equation, including ordinary differential equations and partial differential equations. However, the accuracy and stability of the method may vary depending on the characteristics of the specific equation being solved.

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