class AdsorbingSurfactantEquation():
r"""
The `AdsorbingSurfactantEquation` object solves the
`SurfactantEquation` but with an adsorbing species from some bulk
value. The equation that describes the surfactant adsorbing is
given by,
.. math::
\dot{\theta} = J v \theta + k c (1 - \theta - \theta_{\text{other}}) - \theta c_{\text{other}} k_{\text{other}} - k^- \theta
where :math:`\theta`, :math:`J`, :math:`v`, :math:`k`, :math:`c`,
:math:`k^-` and :math:`n` represent the surfactant coverage, the curvature,
the interface normal velocity, the adsorption rate, the concentration in the
bulk at the interface, the consumption rate and an exponent of consumption,
respectively. The :math:`\text{other}` subscript refers to another
surfactant with greater surface affinity.
The terms on the RHS of the above equation represent conservation of
surfactant on a non-uniform surface, Langmuir adsorption, removal of
surfactant due to adsorption of the other surfactant onto non-vacant sites
and consumption of the surfactant respectively. The adsorption term is added
to the source by setting :math:` S_c = k c (1 - \theta_{\text{other}})` and
:math:`S_p = -k c`. The other terms are added to the source in a similar
way.
The following is a test case:
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> from fipy.tools import numerix
>>> from fipy.variables.cellVariable import CellVariable
>>> dx = .5
>>> dy = 2.3
>>> dt = 0.25
>>> k = 0.56
>>> initialValue = 0.1
>>> c = 0.2
>>> from fipy.meshes import Grid2D
>>> from fipy import serialComm
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = (-dx*3/2, -dx/2, dx/2,
... 3*dx/2, 5*dx/2),
... hasOld = 1)
>>> surfactantVar = SurfactantVariable(value = (0, 0, initialValue, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar = CellVariable(mesh = mesh, value = (c , c, c, c, c))
>>> eqn = AdsorbingSurfactantEquation(surfactantVar = surfactantVar,
... distanceVar = distanceVar,
... bulkVar = bulkVar,
... rateConstant = k)
>>> eqn.solve(surfactantVar, dt = dt)
>>> answer = (initialValue + dt * k * c) / (1 + dt * k * c)
>>> print numerix.allclose(surfactantVar.interfaceVar,
... numerix.array((0, 0, answer, 0, 0)))
1
The following test case is for two surfactant variables. One has more
surface affinity than the other.
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> dx = 0.5
>>> dy = 2.73
>>> dt = 0.001
>>> k0 = 1.
>>> k1 = 10.
>>> theta0 = 0.
>>> theta1 = 0.
>>> c0 = 1.
>>> c1 = 1.
>>> totalSteps = 10
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = dx * (numerix.arange(5) - 1.5),
... hasOld = 1)
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> var1 = SurfactantVariable(value = (0, 0, theta1, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> bulkVar1 = CellVariable(mesh = mesh, value = (c1, c1, c1, c1, c1))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = k0)
>>> eqn1 = AdsorbingSurfactantEquation(surfactantVar = var1,
... distanceVar = distanceVar,
... bulkVar = bulkVar1,
... rateConstant = k1,
... otherVar = var0,
... otherBulkVar = bulkVar0,
... otherRateConstant = k0)
>>> for step in range(totalSteps):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> answer0 = 1 - numerix.exp(-k0 * c0 * dt * totalSteps)
>>> answer1 = (1 - numerix.exp(-k1 * c1 * dt * totalSteps)) * (1 - answer0)
>>> print numerix.allclose(var0.interfaceVar,
... numerix.array((0, 0, answer0, 0, 0)), rtol = 1e-2)
1
>>> print numerix.allclose(var1.interfaceVar,
... numerix.array((0, 0, answer1, 0, 0)), rtol = 1e-2)
1
>>> dt = 0.1
>>> for step in range(10):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> x, y = mesh.cellCenters
>>> check = var0.interfaceVar + var1.interfaceVar
>>> answer = CellVariable(mesh=mesh, value=check)
>>> answer[x==1.25] = 1.
>>> print check.allequal(answer)
True
The following test case is to fix a bug where setting the adsorption
coefficient to zero leads to the solver not converging and an eventual
failure.
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = 0)
>>> eqn0.solve(var0, dt = dt)
>>> eqn0.solve(var0, dt = dt)
>>> answer = CellVariable(mesh=mesh, value=var0.interfaceVar)
>>> answer[x==1.25] = 0.
>>> print var0.interfaceVar.allclose(answer)
True
The following test case is to fix a bug that allows the accelerator to
become negative.
>>> nx = 5
>>> ny = 5
>>> dx = 1.
>>> dy = 1.
>>> mesh = Grid2D(dx=dx, dy=dy, nx = nx, ny = ny, communicator=serialComm)
>>> x, y = mesh.cellCenters
>>> disVar = DistanceVariable(mesh=mesh, value=1., hasOld=True)
>>> disVar[y < dy] = -1
>>> disVar[x < dx] = -1
>>> disVar.calcDistanceFunction() #doctest: +LSM
>>> levVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> accVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> levEq = AdsorbingSurfactantEquation(levVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0)
>>> accEq = AdsorbingSurfactantEquation(accVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0,
... otherVar = levVar,
... otherBulkVar = 0,
... otherRateConstant = 0)
>>> extVar = CellVariable(mesh = mesh, value = accVar.interfaceVar)
>>> from fipy import TransientTerm, AdvectionTerm
>>> advEq = TransientTerm() + AdvectionTerm(extVar)
>>> dt = 0.1
>>> for i in range(50):
... disVar.calcDistanceFunction()
... extVar.value = (numerix.array(accVar.interfaceVar))
... disVar.extendVariable(extVar)
... disVar.updateOld()
... advEq.solve(disVar, dt = dt)
... levEq.solve(levVar, dt = dt)
... accEq.solve(accVar, dt = dt) #doctest: +LSM
>>> # The following test fails sometimes on linux with scipy solvers
>>> # See issue #575. We ignore for now.
>>> print (accVar >= -1e-10).all() #doctest: +NOTLINUXSCIPY
True
"""
def __init__(self,
surfactantVar=None,
distanceVar=None,
bulkVar=None,
rateConstant=None,
otherVar=None,
otherBulkVar=None,
otherRateConstant=None,
consumptionCoeff=None):
"""
Create a `AdsorbingSurfactantEquation` object.
:Parameters:
- `surfactantVar`: The `SurfactantVariable` to be solved for.
- `distanceVar`: The `DistanceVariable` that marks the interface.
- `bulkVar`: The value of the `surfactantVar` in the bulk.
- `rateConstant`: The adsorption rate of the `surfactantVar`.
- `otherVar`: Another `SurfactantVariable` with more surface affinity.
- `otherBulkVar`: The value of the `otherVar` in the bulk.
- `otherRateConstant`: The adsorption rate of the `otherVar`.
- `consumptionCoeff`: The rate that the `surfactantVar` is consumed during deposition.
"""
self.eq = TransientTerm(coeff=1) - ExplicitUpwindConvectionTerm(
SurfactantConvectionVariable(distanceVar))
self.dt = Variable(0.)
mesh = distanceVar.mesh
adsorptionCoeff = self.dt * bulkVar * rateConstant
spCoeff = adsorptionCoeff * distanceVar._cellInterfaceFlag
scCoeff = adsorptionCoeff * distanceVar.cellInterfaceAreas / mesh.cellVolumes
self.eq += ImplicitSourceTerm(spCoeff) - scCoeff
if otherVar is not None:
otherSpCoeff = self.dt * otherBulkVar * otherRateConstant * distanceVar._cellInterfaceFlag
otherScCoeff = -otherVar.interfaceVar * scCoeff
self.eq += ImplicitSourceTerm(otherSpCoeff) - otherScCoeff
vars = (surfactantVar, otherVar)
else:
vars = (surfactantVar, )
total = 0
for var in vars:
total += var.interfaceVar
maxVar = (total > 1) * distanceVar._cellInterfaceFlag
val = distanceVar.cellInterfaceAreas / mesh.cellVolumes
for var in vars[1:]:
val -= distanceVar._cellInterfaceFlag * var
spMaxCoeff = 1e20 * maxVar
scMaxCoeff = spMaxCoeff * val * (val > 0)
self.eq += ImplicitSourceTerm(spMaxCoeff) - scMaxCoeff - 1e-40
if consumptionCoeff is not None:
self.eq += ImplicitSourceTerm(consumptionCoeff)
def solve(self, var, boundaryConditions=(), solver=None, dt=None):
"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear system once.
:Parameters:
- `var`: A `SurfactantVariable` to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
"""
self.dt.setValue(dt)
if solver is None:
import fipy.solvers.solver
if fipy.solvers.solver == 'pyamg':
from fipy.solvers.pyAMG.linearGeneralSolver import LinearGeneralSolver
solver = LinearGeneralSolver(tolerance=1e-15, iterations=2000)
else:
from fipy.solvers import LinearPCGSolver
solver = LinearPCGSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions, )
var.constrain(0, var.mesh.exteriorFaces)
self.eq.solve(var,
boundaryConditions=boundaryConditions,
solver=solver,
dt=1.)
def sweep(self,
var,
solver=None,
boundaryConditions=(),
dt=None,
underRelaxation=None,
residualFn=None):
r"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear
system once. This method also recalculates and returns the
residual as well as applying under-relaxation.
:Parameters:
- `var`: The variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
- `underRelaxation`: Usually a value between `0` and `1` or `None` in the case of no under-relaxation
"""
self.dt.setValue(dt)
if solver is None:
from fipy.solvers import DefaultAsymmetricSolver
solver = DefaultAsymmetricSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions, )
var.constrain(0, var.mesh.exteriorFaces)
return self.eq.sweep(var,
solver=solver,
boundaryConditions=boundaryConditions,
underRelaxation=underRelaxation,
residualFn=residualFn,
dt=1.)
class AdsorbingSurfactantEquation():
r"""
The `AdsorbingSurfactantEquation` object solves the
`SurfactantEquation` but with an adsorbing species from some bulk
value. The equation that describes the surfactant adsorbing is
given by,
.. math::
\dot{\theta} = J v \theta + k c (1 - \theta - \theta_{\text{other}}) - \theta c_{\text{other}} k_{\text{other}} - k^- \theta
where :math:`\theta`, :math:`J`, :math:`v`, :math:`k`, :math:`c`,
:math:`k^-` and :math:`n` represent the surfactant coverage, the curvature,
the interface normal velocity, the adsorption rate, the concentration in the
bulk at the interface, the consumption rate and an exponent of consumption,
respectively. The :math:`\text{other}` subscript refers to another
surfactant with greater surface affinity.
The terms on the RHS of the above equation represent conservation of
surfactant on a non-uniform surface, Langmuir adsorption, removal of
surfactant due to adsorption of the other surfactant onto non-vacant sites
and consumption of the surfactant respectively. The adsorption term is added
to the source by setting :math:` S_c = k c (1 - \theta_{\text{other}})` and
:math:`S_p = -k c`. The other terms are added to the source in a similar
way.
The following is a test case:
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> from fipy.tools import numerix
>>> from fipy.variables.cellVariable import CellVariable
>>> dx = .5
>>> dy = 2.3
>>> dt = 0.25
>>> k = 0.56
>>> initialValue = 0.1
>>> c = 0.2
>>> from fipy.meshes import Grid2D
>>> from fipy import serialComm
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = (-dx*3/2, -dx/2, dx/2,
... 3*dx/2, 5*dx/2),
... hasOld = 1)
>>> surfactantVar = SurfactantVariable(value = (0, 0, initialValue, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar = CellVariable(mesh = mesh, value = (c , c, c, c, c))
>>> eqn = AdsorbingSurfactantEquation(surfactantVar = surfactantVar,
... distanceVar = distanceVar,
... bulkVar = bulkVar,
... rateConstant = k)
>>> eqn.solve(surfactantVar, dt = dt)
>>> answer = (initialValue + dt * k * c) / (1 + dt * k * c)
>>> print numerix.allclose(surfactantVar.interfaceVar,
... numerix.array((0, 0, answer, 0, 0)))
1
The following test case is for two surfactant variables. One has more
surface affinity than the other.
>>> from fipy.variables.distanceVariable \
... import DistanceVariable
>>> from fipy import SurfactantVariable
>>> from fipy.meshes import Grid2D
>>> dx = 0.5
>>> dy = 2.73
>>> dt = 0.001
>>> k0 = 1.
>>> k1 = 10.
>>> theta0 = 0.
>>> theta1 = 0.
>>> c0 = 1.
>>> c1 = 1.
>>> totalSteps = 10
>>> mesh = Grid2D(dx = dx, dy = dy, nx = 5, ny = 1, communicator=serialComm)
>>> distanceVar = DistanceVariable(mesh = mesh,
... value = dx * (numerix.arange(5) - 1.5),
... hasOld = 1)
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> var1 = SurfactantVariable(value = (0, 0, theta1, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> bulkVar1 = CellVariable(mesh = mesh, value = (c1, c1, c1, c1, c1))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = k0)
>>> eqn1 = AdsorbingSurfactantEquation(surfactantVar = var1,
... distanceVar = distanceVar,
... bulkVar = bulkVar1,
... rateConstant = k1,
... otherVar = var0,
... otherBulkVar = bulkVar0,
... otherRateConstant = k0)
>>> for step in range(totalSteps):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> answer0 = 1 - numerix.exp(-k0 * c0 * dt * totalSteps)
>>> answer1 = (1 - numerix.exp(-k1 * c1 * dt * totalSteps)) * (1 - answer0)
>>> print numerix.allclose(var0.interfaceVar,
... numerix.array((0, 0, answer0, 0, 0)), rtol = 1e-2)
1
>>> print numerix.allclose(var1.interfaceVar,
... numerix.array((0, 0, answer1, 0, 0)), rtol = 1e-2)
1
>>> dt = 0.1
>>> for step in range(10):
... eqn0.solve(var0, dt = dt)
... eqn1.solve(var1, dt = dt)
>>> x, y = mesh.cellCenters
>>> check = var0.interfaceVar + var1.interfaceVar
>>> answer = CellVariable(mesh=mesh, value=check)
>>> answer[x==1.25] = 1.
>>> print check.allequal(answer)
True
The following test case is to fix a bug where setting the adosrbtion
coefficient to zero leads to the solver not converging and an eventual
failure.
>>> var0 = SurfactantVariable(value = (0, 0, theta0, 0 ,0),
... distanceVar = distanceVar)
>>> bulkVar0 = CellVariable(mesh = mesh, value = (c0, c0, c0, c0, c0))
>>> eqn0 = AdsorbingSurfactantEquation(surfactantVar = var0,
... distanceVar = distanceVar,
... bulkVar = bulkVar0,
... rateConstant = 0)
>>> eqn0.solve(var0, dt = dt)
>>> eqn0.solve(var0, dt = dt)
>>> answer = CellVariable(mesh=mesh, value=var0.interfaceVar)
>>> answer[x==1.25] = 0.
>>> print var0.interfaceVar.allclose(answer)
True
The following test case is to fix a bug that allows the accelerator to
become negative.
>>> nx = 5
>>> ny = 5
>>> dx = 1.
>>> dy = 1.
>>> mesh = Grid2D(dx=dx, dy=dy, nx = nx, ny = ny, communicator=serialComm)
>>> x, y = mesh.cellCenters
>>> disVar = DistanceVariable(mesh=mesh, value=1., hasOld=True)
>>> disVar[y < dy] = -1
>>> disVar[x < dx] = -1
>>> disVar.calcDistanceFunction() #doctest: +LSM
>>> levVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> accVar = SurfactantVariable(value = 0.5, distanceVar = disVar)
>>> levEq = AdsorbingSurfactantEquation(levVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0)
>>> accEq = AdsorbingSurfactantEquation(accVar,
... distanceVar = disVar,
... bulkVar = 0,
... rateConstant = 0,
... otherVar = levVar,
... otherBulkVar = 0,
... otherRateConstant = 0)
>>> extVar = CellVariable(mesh = mesh, value = accVar.interfaceVar)
>>> from fipy import TransientTerm, AdvectionTerm
>>> advEq = TransientTerm() + AdvectionTerm(extVar)
>>> dt = 0.1
>>> for i in range(50):
... disVar.calcDistanceFunction()
... extVar.value = (numerix.array(accVar.interfaceVar))
... disVar.extendVariable(extVar)
... disVar.updateOld()
... advEq.solve(disVar, dt = dt)
... levEq.solve(levVar, dt = dt)
... accEq.solve(accVar, dt = dt) #doctest: +LSM
>>> print (accVar >= -1e-10).all()
True
"""
def __init__(self,
surfactantVar = None,
distanceVar = None,
bulkVar = None,
rateConstant = None,
otherVar = None,
otherBulkVar = None,
otherRateConstant = None,
consumptionCoeff = None):
"""
Create a `AdsorbingSurfactantEquation` object.
:Parameters:
- `surfactantVar`: The `SurfactantVariable` to be solved for.
- `distanceVar`: The `DistanceVariable` that marks the interface.
- `bulkVar`: The value of the `surfactantVar` in the bulk.
- `rateConstant`: The adsorption rate of the `surfactantVar`.
- `otherVar`: Another `SurfactantVariable` with more surface affinity.
- `otherBulkVar`: The value of the `otherVar` in the bulk.
- `otherRateConstant`: The adsorption rate of the `otherVar`.
- `consumptionCoeff`: The rate that the `surfactantVar` is consumed during deposition.
"""
self.eq = TransientTerm(coeff = 1) - ExplicitUpwindConvectionTerm(SurfactantConvectionVariable(distanceVar))
self.dt = Variable(0.)
mesh = distanceVar.mesh
adsorptionCoeff = self.dt * bulkVar * rateConstant
spCoeff = adsorptionCoeff * distanceVar._cellInterfaceFlag
scCoeff = adsorptionCoeff * distanceVar.cellInterfaceAreas / mesh.cellVolumes
self.eq += ImplicitSourceTerm(spCoeff) - scCoeff
if otherVar is not None:
otherSpCoeff = self.dt * otherBulkVar * otherRateConstant * distanceVar._cellInterfaceFlag
otherScCoeff = -otherVar.interfaceVar * scCoeff
self.eq += ImplicitSourceTerm(otherSpCoeff) - otherScCoeff
vars = (surfactantVar, otherVar)
else:
vars = (surfactantVar,)
total = 0
for var in vars:
total += var.interfaceVar
maxVar = (total > 1) * distanceVar._cellInterfaceFlag
val = distanceVar.cellInterfaceAreas / mesh.cellVolumes
for var in vars[1:]:
val -= distanceVar._cellInterfaceFlag * var
spMaxCoeff = 1e20 * maxVar
scMaxCoeff = spMaxCoeff * val * (val > 0)
self.eq += ImplicitSourceTerm(spMaxCoeff) - scMaxCoeff - 1e-40
if consumptionCoeff is not None:
self.eq += ImplicitSourceTerm(consumptionCoeff)
def solve(self, var, boundaryConditions=(), solver=None, dt=None):
"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear system once.
:Parameters:
- `var`: A `SurfactantVariable` to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
"""
self.dt.setValue(dt)
if solver is None:
import fipy.solvers.solver
if fipy.solvers.solver == 'pyamg':
from fipy.solvers.pyAMG.linearGeneralSolver import LinearGeneralSolver
solver = LinearGeneralSolver(tolerance=1e-15, iterations=2000)
else:
from fipy.solvers import LinearPCGSolver
solver = LinearPCGSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions,)
var.constrain(0, var.mesh.exteriorFaces)
self.eq.solve(var,
boundaryConditions=boundaryConditions,
solver = solver,
dt=1.)
def sweep(self, var, solver=None, boundaryConditions=(), dt=None, underRelaxation=None, residualFn=None):
r"""
Builds and solves the `AdsorbingSurfactantEquation`'s linear
system once. This method also recalculates and returns the
residual as well as applying under-relaxation.
:Parameters:
- `var`: The variable to be solved for. Provides the initial condition, the old value and holds the solution on completion.
- `solver`: The iterative solver to be used to solve the linear system of equations.
- `boundaryConditions`: A tuple of boundaryConditions.
- `dt`: The time step size.
- `underRelaxation`: Usually a value between `0` and `1` or `None` in the case of no under-relaxation
"""
self.dt.setValue(dt)
if solver is None:
from fipy.solvers import DefaultAsymmetricSolver
solver = DefaultAsymmetricSolver()
if type(boundaryConditions) not in (type(()), type([])):
boundaryConditions = (boundaryConditions,)
var.constrain(0, var.mesh.exteriorFaces)
return self.eq.sweep(var, solver=solver, boundaryConditions=boundaryConditions, underRelaxation=underRelaxation, residualFn=residualFn, dt=1.)
