Understanding adaptive timestepping¶
It is helpful to keep in mind this fundamental fact:
length of time steps taken by a model affects results of a simulation.
This applies to all evolutionary models and PISM is no different.
We expect model results to converge to the solution of the continuum problem corresponding to a model as \(\dt\) goes to zero. Also, results using different \(\dt\) should be “close” to this solution and to each other, but they need not be the same.
One important consequence is that changes in PISM settings that may not seem to be related to physical choices may affect results if they affect time stepping.
Here is a possiblyincomplete list of PISM components and settings that may affect time step length.
Numerical stability criteria.
Diffusivitybased time step restriction for the mass continuity (mass transport) step using SIA diffusivity (or its estimate when the Blatter solver is used).
The value of
time_stepping
.adaptive_ratio
adjusting the diffusivitybased time step restriction (see (4)).CFL time step restriction for the mass continuity step using sliding velocity, (or verticallyaveraged horizontal velocity with the Blatter solver).
CFL time step restriction for horizontal advection within the ice volume within energy balance and age models. Uses horizontal (\(u,v\)) components of the ice velocity within the 3D volume of the ice.
Reporting.
If
time_stepping
.hit_ts_times
is set, PISM will adjust time step lengths to “hit” times requested withoutput
.timeseries
.times
.If
time_stepping
.hit_extra_times
is set (the default), PISM will adjust time step lengths to “hit” times requested withoutput
.extra
.times
.If
time_stepping
.hit_save_times
is set, PISM will adjust time step lengths to “hit” times requested withoutput
.snapshot
.times
.
Timestep “skipping” to reduce computational costs:
time_stepping
.skip
.enabled
andtime_stepping
.skip
.max
.This mechanism enables PISM to take a number of “cheap” massbalance steps (including SIA diffusivity updates) before more expensive temperature, age, and SSA stress balance computations are done.
Time step “skipping” roughly corresponds to asynchronous coupling between
ice flow by shear in planes parallel to the geoid and
membranetype ice flow and advection of energy and tracers (such as age).
This is only effective if the time step is being limited by the diffusivity time step restriction associated to mass continuity using the SIA.
PISM computes time step restrictions \(\{\dt_0, \dt_1, \dots, \dt_n \}\) from all of PISM’s submodules and sorts them from from smallest to largest. Then the maximum allowed time step is
\[\dt_{\text{max}} = \dt_0.\]If
time_stepping
.skip
.enabled
is set and the most severe restriction comes from the SIAdiffusivitybased stability criterion for mass continuity, it skips(41)¶\[N = \min\left(\left\lfloor 0.95\, \frac{\dt_1}{\dt_0} \right\rfloor, N_{\text{max}} \right)\]energy, age, and 3D velocity updates, where \(N_{\text{max}}\) is set using
time_stepping
.skip
.max
.Warning
The effects of this mechanism are not well understood. Please use with caution.
The maximum recommended value for
time_stepping
.skip
.max
depends on the context. The temperature field should be updated when the surface changes significantly, and likewise the basal sliding velocity if it comes from the SSA calculation.Atmosphere, surface process, ocean, and sea level forcing components.
The LingleClark bed deformation model (see LingleClark and
bed_deformation
.lc
.update_interval
).If
geometry
.front_retreat
.use_cfl
is set, PISM adjusts time step lengths to satisfy the CFL condition that uses the total front retreat rate coming from calving and frontal melt models.The time step length never exceeds
time_stepping
.maximum_time_step
.If
time_stepping
.hit_multiples
is set to a positive number, PISM will “hit” multiples of the number of model years specified. For example, if stability criteria require a timestep of 11 years and thetimestep_hit_multiples 3
option is set, PISM will take a 9 model year long time step. This can be useful to enforce consistent sampling of periodic climate data.If the value \(R\) set using
time_stepping
.resolution
is positive PISM reduces the time step length so that(42)¶\[\dt = N\cdot R\]for some integer \(N\).
The default \(R\) (\(1\) second) allows PISM to represent model time more accurately, reducing the influence of rounding errors.
Note
This is an intermediateterm solution for an issue reported by Thomas Kleiner: some simulations produced different results with identical inputs but different start years.
We tracked it down to the fact that these simulations ended up using slightly different time step lengths. This, in turn, was caused by differences in the absolute precision of the C++ type
double
for numbers of different magnitudes.The time step length never exceeds the total length of the run.
At each time step the PISM standard output includes “flags” and then a summary of the model state using a few numbers. A typical example is
v$Eh diffusivity (dt=0.83945 in 2 substeps; av dt_sub_mass_cont=0.41972)
S 124791.571: 3.11640 2.25720 3.62041 18099.93737
y SSA: 3 outer iterations, ~17.0 KSP iterations each
The characters “v$Eh
” at the beginning of the flags line, the first line in the above
example, give a very terse description of which physical processes were modeled in that
time step. Here “v
” means that a stress balance was solved to compute the velocity.
Then the enthalpy was updated (“E
”) and the ice thickness and surface elevation were
updated (“h
”). The rest of the line looks like
diffusivity (dt=0.83945 in 2 substeps; av dt_sub_mass_cont=0.41972)
Recall that the PISM time step is determined by an adaptive mechanism. Stable mass
conservation and conservation of energy solutions require such an adaptive timestepping
scheme [30]. The first word we see here, namely “diffusivity
”, is the
adaptivetimestepping “reason”. See Table 24. We also see that
there was a major time step of \(0.83945\) model years divided into \(2\) substeps of about
\(0.42\) years. The parameter time_stepping
.skip
.enabled
enables this mechanism,
while time_stepping
.skip
.max
sets the maximum number of such substeps. The
adaptive mechanism may choose to take fewer substeps than time_stepping
.skip
.max
so as to satisfy certain numerical stability criteria, however.
The second line in the above, the line which starts with “S
”, is the summary. Its
format, and the units for these numbers, is simple and is given by a couple of lines
printed near the beginning of the standard output for the run:
P YEAR: ivol iarea max_diffusivity max_sliding_vel
U years 10^6_km^3 10^6_km^2 m^2 s^1 m/year
That is, in each summary we have the total ice volume, total ice area, maximum diffusivity (of the SIA mass conservation equation), and “maximum sliding velocity”. Specifically, the last number is \(\max(\max(u), \max(v))\), i.e. the number that appears in the CFL time step restriction for the “advective” part of the flow in the mass continuity equation.
Note
max_sliding_vel
reported here is not the same as the maximum sliding speed,
\(\max(\sqrt{u^2 + v^2})\).
The third line of the above example shows that the SSA stress balance was solved. Information on the number of nonlinear (outer) and linear (inner) iterations is provided [29].
PISM output 
Active adaptive constraint or PISM subsystem that limited timestep size 


CFL condition for the “advective” part of the mass continuity equation. Uses (\(u\), \(v\)) components of the verticallyaveraged horizontal ice velocity with the Blatter stress balance model. Uses sliding velocity with all other stress balance choices [29]. 

SIAdiffusivitybased time step restriction for the mass continuity equation [30], [123] 

CFL condition using horizontal (\(u\), \(v\)) components of the ice velocity within the ice volume. Restricts the time step of enthalpy, temperature, or age advection [30]. (This CFL condition does not use the vertical (\(w\)) component of ice velocity: PISM’s 3D advection scheme is explicit in \(x\) and \(y\) and implicit in \(z\).) 

end of prescribed run time 

maximum allowed \(\dt\) set with 

the 

the 

a surface or an atmosphere model 

an ocean model 

a hydrology model stability criterion, see section Subglacial hydrology 

CFL condition using the 2D horizontal retreat rate such as the eigencalving model, see section Calving and front retreat 
Parameters¶
Prefix: time_stepping.
adaptive_ratio
(0.12) Adaptive time stepping ratio for the explicit scheme for the mass balance equation; [30], inequality (25)assume_bed_elevation_changed
(no) If set, assume that bed elevation changes every time step. If not set, PISM tries to avoid unnecessary computations if the bed deformation model did not update bed elevation (to reduce the computational cost a little bit).count_steps
(no) If yes, IceModel::run() will count the number of time steps it took. Sometimes useful for performance evaluation. Counts all steps, regardless of whether processes (mass continuity, energy, velocity, …) occurred within the step.hit_extra_times
(yes) Modify the timestepping mechanism to hit times requested usingoutput
.extra
.times
.hit_multiples
(0 years) Hit every X years, where X is specified using this parameter. Use 0 to disable.hit_save_times
(no) Modify the timestepping mechanism to hit times requested usingoutput
.snapshot
.times
.hit_ts_times
(no) Modify the timestepping mechanism to hit times requested usingoutput
.timeseries
.times
.maximum_time_step
(60 365days) Maximum allowed time step lengthresolution
(1 seconds) Time steps are rounded down to be a multiple of this number (set to zero to allow arbitrary time step lengths)skip
.enabled
(no) Use the temperature, age, and SSA stress balance computation skipping mechanism.skip
.max
(10) Number of massbalance steps, including SIA diffusivity updates, to perform before a the temperature, age, and SSA stress balance computations are done
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