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.