Notes on the timestep in the LISFLOOD algorithm
Some relevant notes on the timestepping from the Coulthard et al., (2013) paper:
However, storage cell flow models of the type described above could develop a ‘chequerboard’ patterning unless a model time step orders of magnitude smaller than that required to maintain the CFL condition was used. This patterning devel- oped where too much water was moved from one cell to the adjacent one (even emptying it), then in the next model iteration the situation would reverse and too much water from the adjacent cell would drain back again. This issue became more apparent when storage cell models were applied at smaller spatial resolutions because it was found that the time step required to maintain stability reduces quadratically with decreasing grid cell size (Hunter et al., 2005).
Bates et al. (2010) developed an elegant solution to this speed/resolution issue by integrating a simple inertia term into a storage cell model. This new formulation of LISFLOOD-FP (described in full below) calculates the amount of discharge between cells as a function of both the present water surface slope, depth between cells, friction and the discharge between cells from the previous iteration (representing inertia).
This simple integration of inertia has the effect of damping out the oscillations caused by routing too much water from one cell to another that lead to the chequer-boarding and allows time steps that scale linearly with the grid cell size according to the CFL condition (Bates et al., 2010).
Furthermore, fluxes can still be calculated explicitly with less computational cost per time step than the Manning's equation as the flow equation now includes only a single fractional power term. This important, simple modification to storage cell models means that time steps over which the LISFLOOD-FP flow model can now be run are orders of magnitude larger than those previously possible with storage cell codes and much closer to those used in LEMs. This, coupled with the computational simplicity of the LISFLOOD-FP code presents an ideal opportunity to marry this new flow model with an LEM.
In CAESAR-Lisflood there are two controls on the model time step. First, for the flow model the CFL condition (Equation 3) and second, for the erosion and deposition components (from CAESAR) the time step is determined by the amount of erosion and deposition in a cell (Coulthard et al., 2002; Van de Wiel et al., 2007). Similar in concept to the CFL in CAESAR the time step is adjusted to restrict the amount of erosion and deposition thus limiting elevation changes (affecting the s lope between cells) that can generate model instabilities (again similar to the chequer-board effects described previously).
This leads to situations where the flow model time step is smaller than the erosion/deposition time step (for example during low flows when there is little or no erosion/deposition) and conversely where the erosion/deposition time step is much smaller than the flow model time step (for example during high flow events where there is significant erosion/deposition). Therefore, CAESAR-Lisflood automatically chooses the smaller time step from either component.
However, during low flows, time steps from the flow model can be up to two orders of magnitude smaller than those from the erosion/deposition components. Despite there being relatively little erosion/deposition or changes in flow volumes, depths or velocities, the time step of the combined model would still be governed by Equation 3, significantly increasing model run times. To overcome this, CAESAR-Lisflood measures the difference between hydrological inputs and drainage basin outputs. If the difference is lower than a user prescribed value (a low flow value) then the overall model time step is conditioned by the erosion/deposition model but the flow model by Equation 3.
In summary, during such low flow situations the time period between two rainfall events may be overlooked in the hydrodynamic model (though all the sediment transport is simulated), so the overall model time advances faster than the hydrodynamic model time. This means the next flow event is reached before the hydrodynamic model has simulated all of the intervening period of low flow. A similar separation of flow and morphological model time steps to increase simulation speed is a method that has been used by Lesser et al. (2004) and Crosato et al. (2012).