Time integration

Time integration methods for the numerical solution of Ordinary Differential Equations (ODEs), also called timesteppers, can be of different nature and flavor (e.g., explicit, semi-implicit, single-stage, multi-stage, single-step, multi-step, single-rate, multi-rate, etc). ClimateMachine supports several of them. Before showing the different nature of some of these methods, let us introduce some common notation.

A commonly used notation for Initial Value Problems (IVPs) is:

\[\begin{align} \frac{\mathrm{d} \boldsymbol{q}}{ \mathrm{d} t} &= \mathcal{T}(t, \boldsymbol{q}),\\ \boldsymbol{q}(t_0) &= \boldsymbol{q_0}, \end{align}\]

where $\boldsymbol{q}$ is an unknown function (vector in most of our cases) of time $t$, which we would like to approximate, and at the initial time $t_0$ the corresponding initial value $\boldsymbol{q}_0$ is given.

The given general formulation, is suitable for single-step explicit schemes. Generally, the equation can be represented in the following canonical form:

\[\begin{align} \dot {\boldsymbol{q}} + \mathcal{F}(t, \boldsymbol{q}) &= \mathcal{G}(t, \boldsymbol{q}), \end{align}\]

where we have used $\dot {\boldsymbol{q}} = d \boldsymbol{q} / dt$. We refer to the term $\mathcal{G}$ as the right-hand-side (RHS) or explicit term, and to the spatial terms of $\mathcal{F}$ as the left-hand-side (LHS) or implicit term.

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