Problem Setup

We aim at solving the initial-value problem (ordinary-differential equation):

\[\der{p}{t} = f \left( p, t \right).\]

As depicted, \(f\) is not only a function of \(p\) but also of \(t\) in general. Here, however, we limit ourselves to cases when \(f\) is autonomous:

\[\der{p}{t} = f \left( p \right),\]

which appears frequently in modeling physical phenomena such as Navier-Stokes equations.

We consider its numerical treatment; in particular, we focus on explicit \(N\)-step Runge-Kutta schemes, which are generally formulated as

\[ \begin{align}\begin{aligned}& q^0 = p^n\\& \text{do} \quad k = 0, N - 1\\& \quad q^{k + 1} = p^n + \sum_{j = 0}^k a_{k + 1, j} f \left( q^j \right) \Delta t\\& \text{end do}\\& p^{n + 1} = q^N\end{aligned}\end{align} \]

When implemented naively, this procedure necessitates buffers to store \(q\), \(f \left( q^0 \right), f \left( q^1 \right), \cdots, f \left( q^{N - 1} \right)\), and \(p^n\). Our primary focus here is to reformulate this into

\[ \begin{align}\begin{aligned}& q^0 = p^n\\& \text{do} \quad k = 0, N - 1\\& \quad r^k = f \left( q^k \right) + \beta^k r^{k - 1}\\& \quad q^{k + 1} = q^k + \gamma^k r^k \Delta t\\& \text{end do}\\& p^{n + 1} = q^N\end{aligned}\end{align} \]

such that only two storages (\(q^k\) and \(r^k\)) are to be allocated.

Note that Runge-Kutta methods are conveniently expressed using the Butcher tableau:

\[\begin{split}\begin{array}{c|cc} b_0 & a_{0,0} & a_{0,1} & \cdots & a_{0,N - 1} \\ b_1 & a_{1,0} & a_{1,1} & \cdots & a_{1,N - 1} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ b_{N - 1} & a_{N - 1,0} & a_{N - 1,1} & \cdots & a_{N - 1,N - 1} \\ \hline & a_{N,0} & a_{N,1} & \cdots & a_{N,N - 1} \\ \end{array}\end{split}\]

Our objective is to convert this table into something like

\[\begin{split}\begin{array}{c|cccc} k & 0 & 1 & \cdots & N - 1 \\ \hline \beta^k & TBA & TBA & \cdots & TBA \\ \gamma^k & TBA & TBA & \cdots & TBA \\ \end{array}\end{split}\]