What Does Convergent Mean in Numerical Methods for Differential Equations?

[PumkinMonster]

http://en.wikipedia.org/wiki/Numerical_ordinary_differential_equations

Could someone be nice to explain me the convergent condition as written in wiki ?
I don't have an idea what it really means I am learning the basics only
Thank you

 

[chwala]

convergence in a laymans language means tending towards a certain point for e.g if you consider the sequence 1/n and take values for n you get 1,1/2,1/3...0 the sequence converges to 0 i.e to mean ⅟∞=0...

 

[HallsofIvy]

I think the specific reference you are referring to is

All the methods mentioned above are convergent. In fact, convergence is a condition sine qua non for any numerical scheme.

This section is referring to numerical methods to solve a differential equations by "iterative" methods- finding one function after another hopefully getting closer and closer to a true solution. That is saying that the sequence of functions converges to a true solution. It might be that a sequence of functions given by a particular method converges very slowly to a true solution (so it's not a very useful method) or that, no matter how many terms you take it is still slightly off a true solutions (taking higher and higher terms in Fourier series around a point at which the function is not continuous will always "miss" the function in some neighborhood of that point- the neighborhood gets smaller, but the "error" never goes to 0. See "Gibbs phenomenon")

However, whether a sequence of functions converges slowly or slightly misses a true solution, it might still be a useful method for some purposes. If, on the other hand, the sequence does not converge at all, it can't possibly be a useful method for finding a solution, even approximately! That is why convergence is a "sine qua non" ("without which, not") to even consider an interation method. Without at least convergence, it can not be a good method!