Wiki says that Runge-Kutta is iterative?

• nonequilibrium
In summary, the Runge-Kutta methods are an important family of iterative methods for the approximation of solutions of ordinary differential equations. They are used in numerical analysis to solve problems such as the Navier–Stokes equations.
nonequilibrium
Hi,

As you can seeh here, according to wikipedia, the Runge-Kutta methods are iterative, but I don't see what is iterative about it. As I see it, starting from a certain value of x where you know y, you calculate y at x+h, then at x+2h, etc. Where is the iteration?

first line of wiki page said:
In numerical analysis, the Runge–Kutta methods are an important family of implicit and explicit iterative methods for the approximation of solutions of ordinary differential equations.

You do calculate $y_{n+1}$ and $t_{n+1}$ given $y_n$ and $t_n$? That's iteration.

That's iteration? Oh... I thought iteration meant that each step you correct the previous value (by using the previous value somehow); such a method would also for example contain a cut-off criterion (when one has converged well enough to a certain solution).

mr. vodka said:
Oh... I thought iteration meant that each step you correct the previous value (by using the previous value somehow);

In a sense, that's what R-K methods do within each time step. You start by doing a forward-difference Euler step, then you make some corrections to it. Finally you take a weighted average of all the iterations as the "best" estimate of the solution.

For an R-K method of a given order, you do a fixed number of iterations rather than using a convergence criterion to decide when to stop.

FWIW there are other numerical methods which you would probably think were more "natural" iterative methods, where it is simpler just to do a fixed number of iterations that willl guarantee convergence (from a mathematical analysis of the method) rather than trying to numerically estimate when it has converged. For example if you can prove that 3 iterations is always enough, it may not be worth the extra cost of checking if 2 iterations is enough, but almost always finding that it isn't.

mr. vodka said:
Oh... I thought iteration meant that each step you correct the previous value (by using the previous value somehow); such a method would also for example contain a cut-off criterion (when one has converged well enough to a certain solution).
What the wikipedia article was referring to is that the 2nd stage of an RK integration step depends on the 1st stage, the 3rd stage depends on the first two stages, and so on.

You're appear to be thinking of adaptive integration techniques here. Runge-Kutta techniques can be used for this (as can many others). One way to do this is to run two integrators side by side, decreasing the step size if the two techniques disagree by too much, increasing it if they don't disagree enough. The widely used adaptive RK45 integrator does just this.

A different type of iteration is used in implicit integration. I'll illustrate with Euler's method, the simplest of all RK techniques. Here integration is performed via $x(t+\Delta t) = x(t) + \Delta t \dot x(x(t),t)$. This is a very lousy technique for a number of reasons, one of which is that it is incredibly unstable. Implicit Euler uses the derivative at the end of the interval instead of at the start: $x(t+\Delta t) = x(t) + \Delta t \dot x(x(t+\Delta t),t+\Delta t)$. There's a problem here: The derivative at the end of the interval depends on the (unknown) state at the end of the interval. Implicit techniques typically iterate until the state at the end of the interval converges to within some limit.

1. What is the Runge-Kutta method?

The Runge-Kutta method is a numerical algorithm used to solve ordinary differential equations. It is a popular method due to its accuracy and ease of implementation.

2. How does the Runge-Kutta method work?

The Runge-Kutta method uses a series of steps to approximate the solution to a differential equation. It involves evaluating the slope of the function at different points and using this information to predict the next point on the solution curve.

3. What are the advantages of using the Runge-Kutta method?

The Runge-Kutta method is more accurate than other numerical methods, such as Euler's method, in approximating the solution to a differential equation. It is also relatively simple to implement and can handle a wide range of differential equations.

4. Are there any limitations to the Runge-Kutta method?

While the Runge-Kutta method is accurate, it can be computationally expensive for certain types of differential equations. It also requires knowledge of the slope of the function at different points, which may not always be readily available.

5. How is the Runge-Kutta method related to iteration?

The Runge-Kutta method is a type of iterative process, meaning that it involves repeating a series of steps until a desired level of accuracy is achieved. In this case, the algorithm iteratively refines the approximation of the solution to the differential equation until the desired accuracy is reached.

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