Runge-Kutta

The Runge-Kutta timestepping method is an explicit, multi-stage scheme. It can be used either to timestep the whole dynamical core, or in combination with the Method-of-Lines transport scheme. The Strong Stability Preserving (SSP) variants are explicit multi-stage methods which combine the values computed over multiple stages as a weighted linear sum, in order to provide stability. A simple forward Euler option is also available.

Runge-Kutta timestepping of the whole dynamical core has not been implemented in combination with physics parametrization schemes.

Rather than splitting terms as in the Quasi-Newton schemes, the Runge-Kutta treats the whole equation set for the state vector \(\boldsymbol{X}\) as one:

\[\frac{\partial\boldsymbol{X}}{\partial t} + \mathcal{G}(\boldsymbol{X}) = \boldsymbol{0},\]

where \(\mathcal{G}\) encompasses forcing and transport terms. An explicit Runge-Kutta scheme advances one timestep by evaluating a sequence of \(M\) stages,

\[\begin{split}\begin{aligned} \boldsymbol{X}^{(0)} &= \boldsymbol{X}^{n}, \\ \boldsymbol{X}^{(m)} &= \boldsymbol{X}^{(0)} - \Delta t\sum_{i=1}^{m} c_{m,i}\mathcal{G}[\boldsymbol{X}^{(i-1)}], \qquad \forall m = 1, \ldots, M, \\ \boldsymbol{X}^{n+1} &= \boldsymbol{X}^{(M)}. \end{aligned}\end{split}\]

The scheme is explicit because each stage \(i\) depends only on previous stages so no implicit solve is required. In practice this means a Runge-Kutta step is a sequence of tendency evaluations and weighted sums, with higher-order SSP variants using more stages than Forward Euler to improve stability and accuracy.

In GungHo, this same stage-update pattern is used whether Runge-Kutta is applied to the whole dynamical core, or used inside Method-of-Lines transport where \(\mathcal{G}\) is the transport tendency operator.