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Mixed implicit and explicit timestepping

The goal of this tutorial is to describe how the timestepping of a ClimaLand model is carried out. We will use forward and backward Euler as a demonstration, but higher order methods are available in ClimaTimesteppers.

Explicit vs. implicit stepping

Given a differential equation for a prognostic variable Y

\frac{d Y }{dt} = g(Y, t) + h(Y, t),

an explicit (forward) Euler step would entail

Y(t+\Delta t) = Y(t) + [g(Y(t), t) + h(Y(t),t)] \times \Delta t,

while an implicit (backward) Euler step would entail

Y(t+\Delta t) = Y(t) + [g(Y(t+\Delta t), t) + h(Y(t+\Delta t),t)] \times \Delta t,

which reqires us to solve an implicit equation for Y(t+ Δt). We usually do so using Newton's method, which requires the derivative of the entire right hand side with respect to the variable we are solving for, Y(t+ Δt). This is called the Jacobian.

Sometimes certain terms must be stepped implicit for numerical stability, while others are more slowly varying or stable. In this case, a mixed approach would be

Y(t+\Delta t) = Y(t) + [g(Y(t+\Delta t), t) + h(Y(t),t)] \times \Delta t,

assuming that h is the slow term, and g is the fast term. Note that solving this implicit equation for (Y(t+ Δt) with Newton's method would be similar to that of the fully implicit approach, but with an approximated Jacobian (neglecting ∂h/∂Y).

If our timestepping scheme involves evaluating all right-hand-side tendencies at the current (known) value of a prognostic variable, we refer to that prognostic variable as explicit. If any of the right-hand-side tendencies are evaluated at the next (unknown) value of the prognostic variable, we refer to it as implicit. In the latter case, the Jacobian would include a term like ∂ tendency/∂ variable, even if it is an approximate (not exact) form of the Jacobian.

Implicit and explicit prognostic variables of the land model

We treat two prognostic variables of the soil model (ϑl, ρeint) and the canopy temperature implicitly, and the canopy water content, the soil ice content, and all prognostic variables of the snow model explicitly.

Implicit vs explicit tendencies - not complete as of 4/23/25

Implicit:

  • Vertical contribution of the divergence of the Darcy flux
  • Vertical contribution of the divergence of diffusive heat flux
  • Vertical contribution of the divergence of heat flux due to Darcy flow
  • Canopy temperature (except root extraction of energy)
  • SHF, LHF, evaporation, and sublimation of soil (note that these are explicit in θ_i!)
  • Soil radiation (does not contribute to Jacobian)
  • Subsurface runoff (this is computed in the same function the same time as surface runoff, but does not contribute to Jacobian.)

Explicit

  • Phase changes in soil
  • Root extraction
  • all snow tendencies
  • Darcy flux within the canopy

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