# Source code for imod.mf6.ims

```
import numpy as np
import xarray as xr
from imod.mf6.pkgbase import Package
from imod.schemata import DTypeSchema
[docs]class Solution(Package):
"""
Iterative Model Solution.
The model solution will solve all of the models that are added to it, as
specified in the simulation name file, and will include Numerical Exchanges,
if they are present. The iterative model solution requires specification of
both nonlinear and linear settings.
https://water.usgs.gov/water-resources/software/MODFLOW-6/mf6io_6.0.4.pdf#page=147
Three predifined solutions settings are available: SolutionPresetSimple,
SolutionPresetModerate and SolutionPresetComplex. When using one of the
predefined solutions only the print_option, csv_output, and no_ptc have to
be defined. The default values for each are described below.
Parameters
----------
modelnames: list of str
Which models to solve in this solution. Only models of the same type
(GWF or GWT) should be added to the same solution.
outer_dvclose: float
real value defining the head change criterion for convergence of the
outer (nonlinear) iterations, in units of length. When the maximum
absolute value of the head change at all nodes during an iteration is
less than or equal to outer_dvclose, iteration stops. Commonly,
outer_dvclose equals 0.01.
SolutionPresetSimple: 0.001
SolutionPresetModerate: 0.01
SolutionPresetComplex: 0.1
outer_maximum: int
integer value defining the maximum number of outer (nonlinear)
iterations – that is, calls to the solution routine. For a linear
problem outer_maximum should be 1.
SolutionPresetSimple: 25
SolutionPresetModerate: 50
SolutionPresetComplex: 100
inner_maximum: int
integer value defining the maximum number of inner (linear) iterations.
The number typically depends on the characteristics of the matrix
solution scheme being used. For nonlinear problems, inner_maximum
usually ranges from 60 to 600; a value of 100 will be sufficient for
most linear problems.
SolutionPresetSimple: 50
SolutionPresetModerate: 100
SolutionPresetComplex: 500
inner_dvclose: float
real value defining the head change criterion for convergence of the
inner (linear) iterations, in units of length. When the maximum absolute
value of the head change at all nodes during an iteration is less than
or equal to inner_dvclose, the matrix solver assumes convergence.
Commonly, inner_dvclose is set an order of magnitude less than the
outer_dvclose value.
SolutionPresetSimple: 0.001
SolutionPresetModerate: 0.01
SolutionPresetComplex: 0.1
inner_rclose: float
real value that defines the flow residual tolerance for convergence of
the IMS linear solver and specific flow residual criteria used. This
value represents the maximum allowable residual at any single node.
Value is in units of length cubed per time, and must be consistent with
MODFLOW 6 length and time units. Usually a value of 1.0 × 10−1 is
sufficient for the flow-residual criteria when meters and seconds are
the defined MODFLOW 6 length and time.
SolutionPresetSimple: 0.1
SolutionPresetModerate: 0.1
SolutionPresetComplex: 0.1
linear_acceleration: str
options: {"cg", "bicgstab"}
a keyword that defines the linear acceleration method used by the
default IMS linear solvers. CG - preconditioned conjugate gradient
method. BICGSTAB - preconditioned bi-conjugate gradient stabilized
method.
SolutionPresetSimple: "cg"
SolutionPresetModerate: "bicgstab"
SolutionPresetComplex: "bicgstab"
under_relaxation: str, optional
options: {"None", "simple", "cooley", "bdb"}
is an optional keyword that defines the nonlinear relative_rclose
schemes used. By default under_relaxation is not used.
None - relative_rclose is not used.
simple - Simple relative_rclose scheme with a fixed relaxation factor is
used.
cooley - Cooley relative_rclose scheme is used.
dbd - delta-bar-delta relative_rclose is used.
Note that the relative_rclose schemes are used in conjunction with
problems that use the Newton-Raphson formulation, however, experience
has indicated that the Cooley relative_rclose and damping work well also
for the Picard scheme with the wet/dry options of MODFLOW 6.
Default value: None
SolutionPresetSimple: None
SolutionPresetModerate: "dbd"
SolutionPresetComplex: "dbd"
under_relaxation_theta: float, optional
real value defining the reduction factor for the learning rate
(underrelaxation term) of the delta-bar-delta algorithm. The value of
under relaxation theta is between zero and one. If the change in the
variable (head) is of opposite sign to that of the previous iteration,
the relative_rclose term is reduced by a factor of under relaxation
theta. The value usually ranges from 0.3 to 0.9; a value of 0.7 works
well for most problems. under relaxation theta only needs to be
specified if under relaxation is dbd.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.9
SolutionPresetComplex: 0.8
under_relaxation_kappa: float, optional
real value defining the increment for the learning rate (relative_rclose
term) of the delta-bar-delta algorithm. The value of under relaxation
kappa is between zero and one. If the change in the variable (head) is
of the same sign to that of the previous iteration, the relative_rclose
term is increased by an increment of under_relaxation_kappa. The value
usually ranges from 0.03 to 0.3; a value of 0.1 works well for most
problems. under relaxation kappa only needs to be specified if under
relaxation is dbd.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0001
SolutionPresetComplex: 0.0001
under_relaxation_gamma: float, optional
real value defining the history or memory term factor of the
delta-bardelta algorithm. under relaxation gamma is between zero and 1
but cannot be equal to one. When under relaxation gamma is zero, only
the most recent history (previous iteration value) is maintained. As
under relaxation gamma is increased, past history of iteration changes
has greater influence on the memory term. The memory term is maintained
as an exponential average of past changes. Retaining some past history
can overcome granular behavior in the calculated function surface and
therefore helps to overcome cyclic patterns of nonconvergence. The value
usually ranges from 0.1 to 0.3; a value of 0.2 works well for most
problems. under relaxation gamma only needs to be specified if under
relaxation is not none.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 0.0
under_relaxation_momentum: float, optional
real value defining the fraction of past history changes that is added
as a momentum term to the step change for a nonlinear iteration. The
value of under relaxation momentum is between zero and one. A large
momentum term should only be used when small learning rates are
expected. Small amounts of the momentum term help convergence. The value
usually ranges from 0.0001 to 0.1; a value of 0.001 works well for most
problems. under relaxation momentum only needs to be specified if under
relaxation is dbd.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 0.0
backtracking_number: int, optional
integer value defining the maximum number of backtracking iterations
allowed for residual reduction computations. If backtracking number = 0
then the backtracking iterations are omitted. The value usually ranges
from 2 to 20; a value of 10 works well for most problems.
Default value: None
SolutionPresetSimple: 0
SolutionPresetModerate: 0
SolutionPresetComplex: 20
backtracking_tolerance: float, optional
real value defining the tolerance for residual change that is allowed
for residual reduction computations. backtracking tolerance should not
be less than one to avoid getting stuck in local minima. A large value
serves to check for extreme residual increases, while a low value serves
to control step size more severely. The value usually ranges from 1.0 to
106; a value of 104 works well for most problems but lower values like
1.1 may be required for harder problems. backtracking tolerance only
needs to be specified if backtracking_number is greater than zero.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 1.05
backtracking_reduction_factor: float, optional
real value defining the reduction in step size used for residual
reduction computations. The value of backtracking reduction factor is
between 142 MODFLOW 6 – Description of Input and Output zero and one.
The value usually ranges from 0.1 to 0.3; a value of 0.2 works well for
most problems. backtracking_reduction_factor only needs to be specified
if backtracking number is greater than zero.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 0.1
backtracking_residual_limit: float, optional
real value defining the limit to which the residual is reduced with
backtracking. If the residual is smaller than
backtracking_residual_limit, then further backtracking is not performed.
A value of 100 is suitable for large problems and residual reduction to
smaller values may only slow down computations. backtracking residual
limit only needs to be specified if backtracking_number is greater than
zero.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 0.002
rclose_option: str, optional
options: {"strict", "l2norm_rclose", "relative_rclose"}
an optional keyword that defines the specific flow residual criterion
used.
strict– an optional keyword that is used to specify that inner rclose
represents a infinity-norm (absolute convergence criteria) and that the
head and flow convergence criteria must be met on the first inner
iteration (this criteria is equivalent to the criteria used by the
MODFLOW-2005 PCG package (Hill, 1990)).
l2norm_rclose – an optionalkeyword that is used to specify that inner
rclose represents a l-2 norm closure criteria instead of a infinity-norm
(absolute convergence criteria). When l2norm_rclose is specified, a
reasonable initial inner rclose value is 0.1 times the number of active
cells when meters and seconds are the defined MODFLOW 6 length and time.
relative_rclose – an optional keyword that is used to specify that
inner_rclose represents a relative L-2 Norm reduction closure criteria
instead of a infinity-Norm (absolute convergence criteria). When
relative_rclose is specified, a reasonable initial inner_rclose value is
1.0 × 10−4 and convergence is achieved for a given inner (linear)
iteration when ∆h ≤ inner_dvclose and the current L-2 Norm is ≤ the
product of the relativ_rclose and the initial L-2 Norm for the current
inner (linear) iteration. If rclose_option is not specified, an absolute
residual (infinity-norm) criterion is used.
Default value: None
SolutionPresetSimple: "strict"
SolutionPresetModerate: "strict"
SolutionPresetComplex: "strict"
relaxation_factor: float, optional
optional real value that defines the relaxation factor used by the
incomplete LU factorization preconditioners (MILU(0) and MILUT).
relaxation_factor is unitless and should be greater than or equal to 0.0
and less than or equal to 1.0. relaxation_factor Iterative Model
Solution 143 values of about 1.0 are commonly used, and experience
suggests that convergence can be optimized in some cases with relax
values of 0.97. A relaxation_factor value of 0.0 will result in either
ILU(0) or ILUT preconditioning (depending on the value specified for
preconditioner_levels and/or preconditioner_drop_tolerance). By default,
relaxation_factor is zero.
Default value: None
SolutionPresetSimple: 0.0
SolutionPresetModerate: 0
SolutionPresetComplex: 0.0
preconditioner_levels: int, optional
optional integer value defining the level of fill for ILU decomposition
used in the ILUT and MILUT preconditioners. Higher levels of fill
provide more robustness but also require more memory. For optimal
performance, it is suggested that a large level of fill be applied (7 or
8) with use of a drop tolerance. Specification of a
preconditioner_levels value greater than zero results in use of the ILUT
preconditioner. By default, preconditioner_levels is zero and the
zero-fill incomplete LU factorization preconditioners (ILU(0) and
MILU(0)) are used.
Default value: None
SolutionPresetSimple: 0
SolutionPresetModerate: 0
SolutionPresetComplex: 5
preconditioner_drop_tolerance: float, optional
optional real value that defines the drop tolerance used to drop
preconditioner terms based on the magnitude of matrix entries in the
ILUT and MILUT preconditioners. A value of 10−4 works well for most
problems. By default, preconditioner_drop_tolerance is zero and the
zero-fill incomplete LU factorization preconditioners (ILU(0) and
MILU(0)) are used.
Default value: None
SolutionPresetSimple: 0
SolutionPresetModerate: 0.0
SolutionPresetComplex: 0.0001
number_orthogonalizations: int, optional
optional integer value defining the interval used to explicitly
recalculate the residual of the flow equation using the solver
coefficient matrix, the latest head estimates, and the right hand side.
For problems that benefit from explicit recalculation of the residual, a
number between 4 and 10 is appropriate. By default,
number_orthogonalizations is zero.
Default value: None
SolutionPresetSimple: 0
SolutionPresetModerate: 0
SolutionPresetComplex: 2
scaling_method: str
options: {"None", "diagonal", "l2norm"}
an optional keyword that defines the matrix scaling approach used. By
default, matrix scaling is not applied.
None - no matrix scaling applied.
diagonal - symmetric matrix scaling using the POLCG preconditioner
scaling method in Hill (1992).
l2norm - symmetric matrix scaling using the L2 norm.
Default value: None
SolutionPresetSimple: None
SolutionPresetModerate: None
SolutionPresetComplex: None
reordering_method: str
options: {"None", "rcm", "md"}
an optional keyword that defines the matrix reordering approach used. By
default, matrix reordering is not applied.
None - original ordering.
rcm - reverse Cuthill McKee ordering.
md - minimum degree ordering
Default value: None
SolutionPresetSimple: None
SolutionPresetModerate: None
SolutionPresetComplex: None
print_option: str
options: {"None", "summary", "all"}
is a flag that controls printing of convergence information from the
solver.
None - means print nothing.
summary - means print only the total
number of iterations and nonlinear residual reduction summaries.
all - means print linear matrix solver convergence information to the
solution listing file and model specific linear matrix solver
convergence information to each model listing file in addition to
SUMMARY information.
Default value: "summary"
SolutionPresetSimple: No Default
SolutionPresetModerate: No Default
SolutionPresetComplex: No Default
csv_output: str, optional
False if no csv is to be written for the output, enter str of filename
if csv is to be written.
Default value: False
SolutionPresetSimple: No Default
SolutionPresetModerate: No Default
SolutionPresetComplex: No Default
no_ptc: ({True, False}, optional)
is a flag that is used to disable pseudo-transient continuation (PTC).
Option only applies to steady-state stress periods for models using the
Newton-Raphson formulation. For many problems, PTC can significantly
improve convergence behavior for steady-state simulations, and for this
reason it is active by default. In some cases, however, PTC can worsen
the convergence behavior, especially when the initial conditions are
similar to the solution. When the initial conditions are similar to, or
exactly the same as, the solution and convergence is slow, then this NO
PTC option should be used to deactivate PTC. This NO PTC option should
also be used in order to compare convergence behavior with other MODFLOW
versions, as PTC is only available in MODFLOW 6.
Default value: False
SolutionPresetSimple: No Default
SolutionPresetModerate: No Default
SolutionPresetComplex: No Default
validate: {True, False}
Flag to indicate whether the package should be validated upon
initialization. This raises a ValidationError if package input is
provided in the wrong manner. Defaults to True.
"""
_pkg_id = "ims"
_keyword_map = {}
_init_schemata = {
"outer_dvclose": [DTypeSchema(np.floating)],
"outer_maximum": [DTypeSchema(np.integer)],
"inner_maximum": [DTypeSchema(np.integer)],
"inner_dvclose": [DTypeSchema(np.floating)],
"inner_rclose": [DTypeSchema(np.floating)],
"under_relaxation_theta": [DTypeSchema(np.floating)],
"under_relaxation_kappa": [DTypeSchema(np.floating)],
"under_relaxation_gamma": [DTypeSchema(np.floating)],
"under_relaxation_momentum": [DTypeSchema(np.floating)],
"backtracking_number": [DTypeSchema(np.integer)],
"backtracking_tolerance": [DTypeSchema(np.floating)],
"backtracking_reduction_factor": [DTypeSchema(np.floating)],
"backtracking_residual_limit": [DTypeSchema(np.floating)],
"number_orthogonalizations": [DTypeSchema(np.integer)],
}
_template = Package._initialize_template(_pkg_id)
[docs] def __init__(
self,
modelnames,
outer_dvclose,
outer_maximum,
inner_maximum,
inner_dvclose,
inner_rclose,
linear_acceleration,
under_relaxation=None,
under_relaxation_theta=None,
under_relaxation_kappa=None,
under_relaxation_gamma=None,
under_relaxation_momentum=None,
backtracking_number=None,
backtracking_tolerance=None,
backtracking_reduction_factor=None,
backtracking_residual_limit=None,
rclose_option=None,
relaxation_factor=None,
preconditioner_levels=None,
preconditioner_drop_tolerance=None,
number_orthogonalizations=None,
scaling_method=None,
reordering_method=None,
print_option="summary",
csv_output=False,
no_ptc=False,
validate: bool = True,
):
super().__init__()
self.dataset = xr.Dataset()
# Make sure the modelnames are set as a variable rather than dimension:
if isinstance(modelnames, xr.DataArray):
self.dataset["modelnames"] = modelnames
else:
self.dataset["modelnames"] = ("model", modelnames)
self.dataset["outer_dvclose"] = outer_dvclose
self.dataset["outer_maximum"] = outer_maximum
self.dataset["under_relaxation"] = under_relaxation
self.dataset["under_relaxation_theta"] = under_relaxation_theta
self.dataset["under_relaxation_kappa"] = under_relaxation_kappa
self.dataset["under_relaxation_gamma"] = under_relaxation_gamma
self.dataset["under_relaxation_momentum"] = under_relaxation_momentum
self.dataset["backtracking_number"] = backtracking_number
self.dataset["backtracking_tolerance"] = backtracking_tolerance
self.dataset["backtracking_reduction_factor"] = backtracking_reduction_factor
self.dataset["backtracking_residual_limit"] = backtracking_residual_limit
self.dataset["inner_maximum"] = inner_maximum
self.dataset["inner_dvclose"] = inner_dvclose
self.dataset["inner_rclose"] = inner_rclose
self.dataset["rclose_option"] = rclose_option
self.dataset["linear_acceleration"] = linear_acceleration
self.dataset["relaxation_factor"] = relaxation_factor
self.dataset["preconditioner_levels"] = preconditioner_levels
self.dataset["preconditioner_drop_tolerance"] = preconditioner_drop_tolerance
self.dataset["number_orthogonalizations"] = number_orthogonalizations
self.dataset["scaling_method"] = scaling_method
self.dataset["reordering_method"] = reordering_method
self.dataset["print_option"] = print_option
self.dataset["csv_output"] = csv_output
self.dataset["no_ptc"] = no_ptc
self._validate_init_schemata(validate)
[docs]def SolutionPresetSimple(
modelnames, print_option="summary", csv_output=False, no_ptc=False
):
solution = Solution(
modelnames=modelnames,
print_option=print_option,
csv_output=csv_output,
no_ptc=no_ptc,
outer_dvclose=0.001,
outer_maximum=25,
under_relaxation=None,
under_relaxation_theta=0.0,
under_relaxation_kappa=0.0,
under_relaxation_gamma=0.0,
under_relaxation_momentum=0.0,
backtracking_number=0,
backtracking_tolerance=0.0,
backtracking_reduction_factor=0.0,
backtracking_residual_limit=0.0,
inner_maximum=50,
inner_dvclose=0.001,
inner_rclose=0.1,
rclose_option="strict",
linear_acceleration="cg",
relaxation_factor=0.0,
preconditioner_levels=0,
preconditioner_drop_tolerance=0,
number_orthogonalizations=0,
scaling_method=None,
reordering_method=None,
)
return solution
[docs]def SolutionPresetModerate(
modelnames, print_option="summary", csv_output=False, no_ptc=False
):
solution = Solution(
modelnames=modelnames,
print_option=print_option,
csv_output=csv_output,
no_ptc=no_ptc,
outer_dvclose=0.01,
outer_maximum=50,
under_relaxation="dbd",
under_relaxation_theta=0.9,
under_relaxation_kappa=0.0001,
under_relaxation_gamma=0.0,
under_relaxation_momentum=0.0,
backtracking_number=0,
backtracking_tolerance=0.0,
backtracking_reduction_factor=0.0,
backtracking_residual_limit=0.0,
inner_maximum=100,
inner_dvclose=0.01,
inner_rclose=0.1,
rclose_option="strict",
linear_acceleration="bicgstab",
relaxation_factor=0,
preconditioner_levels=0,
preconditioner_drop_tolerance=0.0,
number_orthogonalizations=0,
scaling_method=None,
reordering_method=None,
)
return solution
[docs]def SolutionPresetComplex(
modelnames, print_option="summary", csv_output=False, no_ptc=False
):
solution = Solution(
modelnames=modelnames,
print_option=print_option,
csv_output=csv_output,
no_ptc=no_ptc,
outer_dvclose=0.1,
outer_maximum=100,
under_relaxation="dbd",
under_relaxation_theta=0.8,
under_relaxation_kappa=0.0001,
under_relaxation_gamma=0.0,
under_relaxation_momentum=0.0,
backtracking_number=20,
backtracking_tolerance=1.05,
backtracking_reduction_factor=0.1,
backtracking_residual_limit=0.002,
inner_maximum=500,
inner_dvclose=0.1,
inner_rclose=0.1,
rclose_option="strict",
linear_acceleration="bicgstab",
relaxation_factor=0.0,
preconditioner_levels=5,
preconditioner_drop_tolerance=0.0001,
number_orthogonalizations=2,
scaling_method=None,
reordering_method=None,
)
return solution
```