Register to reply

Stability of an ODE and Euler's method

by Master J
Tags: euler, method, stability
Share this thread:
Master J
#1
Nov23-12, 10:35 AM
P: 225
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 ???
Phys.Org News Partner Science news on Phys.org
Experts defend operational earthquake forecasting, counter critiques
EU urged to convert TV frequencies to mobile broadband
Sierra Nevada freshwater runoff could drop 26 percent by 2100
haruspex
#2
Nov23-12, 09:48 PM
Homework
Sci Advisor
HW Helper
Thanks
P: 9,873
Does this help http://courses.engr.illinois.edu/cs4...stability.pdf?
AlephZero
#3
Nov24-12, 11:52 AM
Engineering
Sci Advisor
HW Helper
Thanks
P: 7,177
Quote Quote by Master J View Post
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.

Master J
#4
Nov29-12, 09:48 AM
P: 225
Stability of an ODE and Euler's method

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 eachother. 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/cs4...stability.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!!


Register to reply

Related Discussions
Integrating Euler's equations for rigid body dynamics with Euler's Method Differential Equations 0
Stability criteria for explicit method Differential Equations 0
Test Stability using Routh Stability Method Engineering, Comp Sci, & Technology Homework 2
Euler methond and the improved Euler method Differential Equations 4
Differential equations, euler's method and bisection method Calculus & Beyond Homework 3