Stability of an ODE and Euler's method

In summary, Euler's method is stable if h is small, but is otherwise useless for solving ODEs that are unstable.
  • #1
Master J
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I have been thinking about numerical methods for ODEs, and the whole notion of stability confuses me.

Take Euler's method for solving an ODE:

U_n+1 = U_n + h.A.U_n

where U_n = U_n( t ), A is the Jacobian and h is step size.

Rearrange:

U_n+1 = ( 1 + hA ).U_n

This method is only stable if (1 + hA) < 1 ( using the eigenvalues of A). But what does this mean!?? Every value of my function that I am numerically getting is less than the previous value. This seems rather useless, I don't get it? It appears to me that this method can only be used on functions that are strictly decreasing for all increasing t ?
 
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  • #3
Master J said:
This seems rather useless

Yup. Euler's (forward difference) method IS "rather useless". In fact compared with almost any other numerical integration method, its not so much "rather useless" as "completely useless".

But it's a nice example of something that "obviosuly" look like a good idea, but turns out not to be.
 
  • #4
Well, I'm still confused.

Say I have an ODE who's solution family y(t) is unstable. That is, for increasing t, the solution curves diverge from each other. In this case, J = df(y, t)/dy < 0.

So does this mean that ANY numerical method I use to solve this ODE will be unstable? With reference to http://courses.engr.illinois.edu/cs450/sp2010/odestability.pdf? there is a condition for all the methods, even the trapezoid rule etc. to be stable. And in each of these it implies that numerical values for each succesive value of y(t) are less than the previous, ie. y(t+h) < y(t).

So, in essence, what I gather here is that unless an ODE has the property that the magnitude of each value of the function y is LESS than the previous value, then it CANNOT be solved with a numerical method accurately?
Or, in another way, errors will always grow in solving an unstable ODE?

All this seems rather strange to me then. We cannot solve an ODE accurately unless the function is monotonically decreasing? What rather tiny area of applicability then!
 
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  • #5



I understand your confusion about the stability of Euler's method for solving ODEs. Let me provide some clarification on this topic.

Stability in numerical methods refers to the ability of the method to produce accurate and reliable results. In the context of ODEs, stability is particularly important because the solution of the ODE can be very sensitive to small changes in the initial conditions or parameters. A stable method will produce solutions that are close to the true solution, while an unstable method will produce solutions that diverge from the true solution.

In the case of Euler's method, stability is determined by the value of the Jacobian matrix, A, and the step size, h. The Jacobian matrix represents the local behavior of the ODE at a specific point, and its eigenvalues determine the stability of the method. If the eigenvalues of A are all negative, then (1 + hA) will always be less than 1, and the method will be stable. However, if any of the eigenvalues are positive, then (1 + hA) can be greater than 1, and the method will be unstable.

So what does this mean in terms of using Euler's method for solving ODEs? It means that the method will only produce accurate results if the eigenvalues of A are all negative. This is not always the case, and therefore, Euler's method may not be the best choice for solving certain types of ODEs. In those cases, other numerical methods, such as the Runge-Kutta method, may be more suitable.

In summary, the stability of Euler's method is determined by the eigenvalues of the Jacobian matrix, and it is important to understand the behavior of the ODE and the properties of the method before using it to solve a particular problem. I hope this explanation helps to clarify the concept of stability in numerical methods for ODEs.
 

1. What is an ODE?

An ODE, or Ordinary Differential Equation, is a type of mathematical equation that involves one or more independent variables and the derivatives of a dependent variable with respect to those independent variables.

2. Why is the stability of an ODE important?

The stability of an ODE is important because it determines the behavior and accuracy of numerical methods used to solve the equation. A stable ODE will have solutions that do not diverge and can be accurately calculated, while an unstable ODE will have solutions that become increasingly inaccurate or even diverge.

3. How is stability of an ODE determined?

The stability of an ODE can be determined by analyzing the eigenvalues of the matrix associated with the equation. If all eigenvalues have negative real parts, the equation is stable. If any eigenvalues have positive real parts, the equation is unstable.

4. What is Euler's method?

Euler's method is a numerical method for solving ODEs by approximating the solution at discrete points. It uses the derivative of the solution at a given point to estimate the solution at the next point. This process is repeated until the desired solution is reached.

5. How does Euler's method relate to the stability of an ODE?

Euler's method can only produce accurate solutions for stable ODEs. If the ODE is unstable, the approximations from Euler's method will become increasingly inaccurate or even diverge. Therefore, it is important to consider the stability of an ODE before using Euler's method to solve it.

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