9.4 Runge–Kutta methods

Runge–Kutta methods evaluate \(f\) at several points within each step and combine the values so as to reproduce the Taylor method to a given order.

Second-order methods

Look for a method \(w_{i+1} = w_i+h\,a_1f(t_i+\alpha _1,w_i+\beta _1)\) that agrees with \(T^{(2)} = f+\frac h2\left (f_t+f_yf\right )\) up to \(O(h^2)\). By Taylor’s Theorem in two variables, \[a_1f(t+\alpha _1,y+\beta _1) = a_1f+a_1\alpha _1f_t+a_1\beta _1f_y+O(\alpha _1^2+\beta _1^2) .\] Matching coefficients requires \(a_1 = 1\), \(\alpha _1 = \frac h2\) and \(\beta _1 = \frac h2f(t,y)\). This gives the midpoint method \[w_{i+1} = w_i+hf\!\left (t_i+\frac h2,\ w_i+\frac h2f(t_i,w_i)\right ),\] which evaluates the slope at the midpoint of the Euler step. Allowing two terms, \(a_1f(t,y)+a_2f(t+\alpha _2,y+\delta _2f(t,y))\), gives a one-parameter family of second-order methods; the choice \(a_1 = a_2 = \frac 12\), \(\alpha _2 = \delta _2 = h\) is the modified Euler method (also called Heun’s method or the improved Euler method): \[w_{i+1} = w_i+\frac h2\Bigl [f(t_i,w_i)+f\bigl (t_{i+1},\ w_i+hf(t_i,w_i)\bigr )\Bigr ].\] It takes an Euler step to predict \(w_{i+1}\), evaluates the slope there, and averages it with the slope at the start: the trapezoidal rule applied to \(\int _{t_i}^{t_{i+1}}y'\,dt\). Both methods have local truncation error \(O(h^2)\).

The classical fourth-order method

The most widely used method is the Runge–Kutta method of order four: \begin {align*} k_1 &= hf(t_i,w_i),\\ k_2 &= hf\!\left (t_i+\frac h2,\ w_i+\frac 12k_1\right ),\\ k_3 &= hf\!\left (t_i+\frac h2,\ w_i+\frac 12k_2\right ),\\ k_4 &= hf(t_{i+1},\ w_i+k_3),\\ w_{i+1} &= w_i+\frac 16(k_1+2k_2+2k_3+k_4). \end {align*}

It has local truncation error \(O(h^4)\), provided \(y\in C^5\). When \(f\) does not depend on \(y\), it reduces to Simpson’s rule for \(\int _{t_i}^{t_{i+1}}f(t)\,dt\).

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