number of iteration for steady state solution of a hyperbolic equation

Question

I'm solving a hyperbolic PDE to steady state using an iterative method. Is there any way to fix the number of iterations which will ensure the steady state solution?

Answer 1

In all numerical methods there will always be some error. Each iteration usually reduces that error and there will come a point where the error is small enough that you are happy with the values you have achieved.

Suppose you are estimating a single value $X$ using a sequence of estimates $x_0, x_1, x_2, ...$.

The error is of course the difference between the estimate and the true value: $\epsilon_n = x_n - X$.

In first order iterative methods we find that $\epsilon_{n+1}=k\epsilon_{n}$, with $|k|<1$

If we know $\epsilon_n$, we can estimate how many further iterations are needed to achieve a required error $\epsilon_r$ by requiring $\epsilon_n\times k^m\le e_r \Rightarrow m \log k\ge\log \epsilon_r-\log \epsilon_n \Rightarrow m \ge \frac {\log \epsilon_r-\log \epsilon_n}{\log k}$

Of course we may not know the true value $X$, but a good proxy for $\epsilon_n$ is the difference in successive estimates $x_{n+1}-x_n$.

In your case you are not measuring the error of a single valkue but a set of values. You will need to create an error function that is a summary of all the errors. Sum of squared errors is a good choice for this.