# What is the advantage of using a particular RK Scheme?

The Wikipedia article on Runge-Kutta Methods lists several examples of each order. My question is, are there any particular advantages using one particular scheme over another of the same order?

I wanted to complete a convergence test of the Advection Equation, which is as follows: $$\frac{\partial \theta}{\partial t} + a\frac{\partial \theta}{\partial x} = 0$$ I wanted to show the effects of exceeding the CFL Condition, which states that: $$\frac{|a| \Delta t}{\Delta x} \leq \mathrm{CFL}_\mathrm{max}$$ I implemented this by picking an arbitrary $$\mathrm{CFL}_\mathrm{max}$$, finding $$\Delta t$$, integrating the advection equation (in spectral space, assuming periodicity) up until $$t = 10$$, and finding the $$\ell_2$$-norm of the error at the nearest timestep (since it might not line up perfectly), while keeping $$\Delta x$$ constant. As can be seen, every single RK3 method is literally superimposed on one another. So what are the benefits of using one over another?

As an aside, I'm not particularly sure why this we get unstable results at the following boundaries:

• RK1: $$\mathrm{CFL}_\mathrm{max} \approx 0.07$$
• RK2: $$\mathrm{CFL}_\mathrm{max} \approx 0.3$$
• RK3: $$\mathrm{CFL}_\mathrm{max} \approx 0.6$$
• RK4: $$\mathrm{CFL}_\mathrm{max} \approx 0.9$$

Can I find the appropriate $$\mathrm{CFL}_\mathrm{max}$$ for a given method, outside of just experimentally testing?

• There are whole books written about the answer to your question. As for finding the maximum stable step size, just get any introductory textbook and read about absolute stability. I recommend this one: staff.washington.edu/rjl/fdmbook Commented Mar 14 at 11:30
• beautiful plot. Never seen a comparison like this. Commented Mar 15 at 6:35
• I am surprised you see any stability for explicit Euler. Commented Mar 15 at 16:09
• @VladimirFГероямслава, I’m guessing that $t=10$ was just not long enough. I believe that RK3 is the minimum stable explicit scheme for the advection equation. Commented Mar 16 at 14:58
• What did you use for spacial discritizaition? Commented Mar 16 at 19:47

You're looking only at the errors themselves and not other properties of the solution. There are sometimes good cases to consider lower-order schemes because they better preserve important characteristics. For example, suppose that, in solving the advection equation you've written above, it's important to strictly preserve non-negativity of the solution $$\theta$$ because failure to do so could make another part of the overall problem that depends on $$\theta$$ ill-posed. The backward Euler scheme will preserve positivity and thus might be preferable over, say, the midpoint rule, which can introduce oscillations that go outside the initial range of the solution. This is all despite the fact that backward Euler is only 1st-order accurate the midpoint rule is 2nd-order. Another way of saying this is that certain schemes tend to have diffusive errors and others tend to have dispersive errors and which one is more tolerable is application-dependent. You can read more about these distinctions in Durran's book.