The Computable Differential Equation Lecture ... - Bruce E. Shapiro

CHAPTER 5. RUNGE-KUTTA METHODS 99
This gives us a numerical approximation of y(1) ≈ 2.71735, and error of approximately
0.034% (the exact value is e ≈ 2.71828. By comparison, a forward Euler
computation with the same step size will yield a numerical result of 2.25, an error
approximately 17%.
Since it is an explicit method, the Runge-Kutta 4-stage method is very easy to
implement in a computer, even though calculations are very tedious to do by hand.
Here is a Mathematica implementation.
RK4[f , {t0 , y0 }, h , tmax ] := Module[
{k1, k2, k3, k4, t, yval, r},
r = {{t0, y0}};
yval = y0;
For[t = t0, t < tmax, t += h,
k1 = yval;
k2 = yval + (h/2) f[t, k1];
k3 = yval + (h/2)f[t + h/2, k2];
k4 = yval + h f[t + h/2, k3];
yval = yval + (h/6) *(f[t, k1] + 2f[t + h/2, k2] + 2
+ h/2, k3] + f[t + h, k4]);
AppendTo[r, {t + h, yval}];
];
Return[r];
]
f[t
5.4 General Form of Runge-Kutta Methods
The general form of the s-stage Runge-Kutta Method is
s∑
y n = y n−1 + h b i K i (5.95)
i=1
where
⎛
K i = f ⎝t n−1 + c i h, y n−1 + h
s∑
j=1
a ij K j
⎞
⎠ (5.96)
and
c i =
s∑
a ij (5.97)
j=1
The coefficients a ij , b j are chosen by comparing the method with a Taylor series expansion
and choosing values that cancel specific terms in the error. There are many
c○2007, B.E.Shapiro
Last revised: May 23, 2007
Math 582B, Spring 2007
California State University Northridge