10.5: Runge-Kutta Methods

The three methods that we have surveyed thus far (Forward Euler, Backward Euler, and Adams-Moulton) have all involved sampling the derivative function $F(y,t)$ at one of the discrete time steps $\{t_n\}$, and the solutions at those time steps $\{\vec{y}_n\}$. It is natural to ask whether we can improve the accuracy by sampling the derivative function at "intermediate" values of $t$ and $\vec{y}$. This is the basic idea behind a family of numerical methods known as Runge-Kutta methods.

Here is a simple version of the method, known as second-order Runge-Kutta (RK2). Our goal is to replace the derivative term with a pair of terms, of the form

$$\vec{y}_{n+1} = \vec{y}_n + A h \vec{F}_A + B h \vec{F}_B,$$

where

$$\begin{align}\vec{F}_A &= \vec{F}(\vec{y}_n, t_n)\\\vec{F}_B &= \vec{F}(\vec{y}_n + \beta \vec{F}_A, t_n + \alpha).\end{align}$$

The coefficients $\{A, B, \alpha, \beta\}$ are adjustable parameters whose values we'll shortly choose, so as to minimize the local truncation error.

During each time step, we start out knowing the solution $\vec{y}_n$ at time $t_{n}$, we first calculate $\vec{F}_{A}$ (which is the derivative term that goes into the Forward Euler method); then we use that to calculate an "intermediate" derivative term $\vec{F}_{B}$. Finally, we use a weighted average of $\vec{F}_{A}$ and $\vec{F}_{B}$ as the derivative term for calculating $\vec{y}_{n+1}$. From this, it is evident that this is an explicit method: for each of the sub-equations, the "right-hand sides" contain known quantities.

We now have to determine the appropriate values of the parameters $\{A, B, \alpha, \beta\}$. First, we Taylor expand $\vec{y}_{n+1}$ around $t_{n}$, using the chain rule:

$$\begin{align} \vec{y}_{n+1} &= \vec{y}_n + h \left.\frac{d\vec{y}}{dt}\right|_{t_n} + \frac{h^2}{2} \left.\frac{d^2\vec{y}}{dt^2}\right|_{t_n} + O(h^3)\\&= \vec{y}_n + h \vec{F}(\vec{y}_n, t_n) + \frac{h^2}{2} \left[ \frac{d}{dt}\vec{F}(\vec{y}(t), t)\right]_{t_n} + O(h^3) \\&= \vec{y}_n + h \vec{F}(\vec{y}_n, t_n) + \frac{h^2}{2} \left[ \sum_j \frac{\partial \vec{F}}{\partial y_j}\, \frac{dy_j }{dt} + \frac{\partial \vec{F}}{\partial t}\right]_{t_n} + O(h^3) \\&= \vec{y}_n + h \vec{F}_A + \frac{h^2}{2} \left\{ \sum_j \left[\frac{\partial \vec{F}}{\partial y_j} \right]_{t_n} \!\! F_{Aj} \; +\; \left[\frac{\partial \vec{F}}{\partial t}\right]_{t_n}\right\} + O(h^3) \end{align}$$

In the same way, we Taylor expand the intermediate derivative term $F_{B}$, whose formula was given above:

$$F_B = F_A + \beta \sum_j F_{Aj} \left[\frac{\partial F}{\partial y_j}\right]_{t_n} + \alpha \left[\frac{\partial F}{\partial t}\right]_{t_n}.$$

If we compare these Taylor expansions to the RK2 formula, then it can be seen that the terms can be made to match up to (and including) $O(h^{2})$, if the parameters are chosen to obey the equations

$$A + B = 1, \quad \alpha = \beta = \frac{h}{2B}.$$

One possible set of solutions is $A = B = 1/2$ and $\alpha = \beta = h$. With these conditions met, the RK2 method has local truncation error of $O(h^{3})$, one order better than the Forward Euler Method (which is likewise an explicit method), and comparable to the Adams-Moulton Method (which is an implicit method).

The local truncation error can be further reduced by taking more intermediate samples of the derivative function. The most commonly-used Runge-Kutta method is the fourth-order Runge Kutta method (RK4), which is given by

$$\begin{align} \vec{y}_{n+1} &= \vec{y}_n + \frac{h}{6}\left(\vec{F}_A + 2\vec{F}_B + 2\vec{F}_C + \vec{F}_D \right)\\ \vec{F}_A &= \vec{F}(\vec{y}_n,\, t_n), \\ \vec{F}_B &= \vec{F}(\vec{y}_n + \tfrac{h}{2} \vec{F}_A,\, t_n + \tfrac{h}{2}), \\ \vec{F}_C &= \vec{F}(\vec{y}_n + \tfrac{h}{2} \vec{F}_B,\, t_n + \tfrac{h}{2}), \\ \vec{F}_D &= \vec{F}(y_n + h\vec{F}_C,\, t_n + h). \end{align}$$

This has local truncation error of $O(h^{5})$. It is an explicit method, and therefore has the disadvantage of being unstable if the problem is stiff and $h$ is sufficiently large.