6.2, i.e. the other (more improved) (better) numerical solver.

Section 6.2 is entitled “Runge-Kutta Methods.”

This is just another numerical solver for differential equations. This is similar to Euler’s method because this method is also a fixed-step solver. Our step size is still the same as last time: h = (b – a)/N. Like last time, we’ll set t₀ to be a, and t₁ would be t₀ + h (= a + h), etc., until the final value t_N = a + Nh = b. Like Euler’s method, our dependent variable (in this case, y) will be chosen iteratively.

At this point you might be asking, “So what’s the difference between this method and Euler’s?”

To which I answer, “Hold your horses, buckaroo. We’re getting to the best and worst part about this method right now.”

The difference is that it’s more difficult to find a geometric interpretation of this method, as compared to Euler’s. Hooray!

(Let me just point out that the section on Euler’s method is twice as long as our current section. This makes me fear for my life as I move onto the next page.)

So there are different orders to the Runge-Kutta  method. The second-order version of this method is also called the improved Euler’s method. We start with our initial point (t₀, y₀) and we compute two different slopes from it:

If we think back to what we did in Euler’s formula, we see we replaced the slope f(t₀, y₀) with the average of our slopes. This isn’t intuitive or anything on the surface, but Taylor’s formula tells us that the truncation error improves drastically with this replacement. So let’s define all of our values inductively:

So when we compute maximum error from this, we get

Our constant L is the same as last time, but M is not. Note: “…again it depends only on f(t, y). Its exact formula is not too useful” (256).

The power of our step size h (in both Euler’s method and this method) is called the order of the solver. This is where we get the second-order part of the second-order Runge-Kutta method (because the h is squared…). This would make Euler’s method a first-order method.

Our good and bad news about the error is basically the same as Euler’s method. The good news comes from our step size, but it’s even better news now, because we can pick incredibly small steps and BOOM our error can get really low. However, we still have the problem with the interval b – a. This means this method becomes incredibly unreliable over long intervals.

The final topic we shall be covering in this section is the solution method. It’s widely used and fairly fast. It’s also much more accurate than Euler’s method. It’s known as the fourth-order Runge-Kutta method.

So we have three cases now: Euler’s method, the second-order and the fourth-order Runge-Kutta method. For all three of these methods, we see that y_i is found by adding y_i-1 to some sort of average of the slopes. In Euler’s method, the average is just one slope, while the average of the slopes from the Runge-Kutta methods is slightly more difficult.

However, the payoff is a much improved error.

The constant L is the same as before, while M is different from both of our previous methods. Again, it doesn’t matter enough to be listed. We see now why this method is called a fourth-order, since the step size is to the fourth power. Something highly important to note about this method is that it produces the most accurate of results in terms of error.

Okay, that’s it for section 6.2 (I know, it was a pretty fast one this time). The next section shall be in chapter 7, so I’ll meet you there.