def homogenize(bc):
"""
Return a homogeneous version of the given boundary condition.
*Arguments*
bc
a :py:class:`DirichletBC <dolfin.fem.bcs.DirichletBC>` instance,
or a list/tuple of
:py:class:`DirichletBC <dolfin.fem.bcs.DirichletBC>` instances.
Other types of boundary conditions are ignored.
If the given boundary condition is a list of boundary conditions,
then a list of homogeneous boundary conditions is returned.
"""
# Handle case when boundary condition is a list
if isinstance(bc, (list, tuple)):
bcs = bc
return [homogenize(bc) for bc in bcs]
# Only consider Dirichlet boundary conditions
if not isinstance(bc, cpp.DirichletBC):
cpp.dolfin_error("bcs.py", "homogenize boundary condition",
"Can only homogenize DirichletBCs")
# Create zero function
V = bc.function_space()
if V.element().value_rank() == 0:
zero = Constant(0)
elif V.element().value_rank() == 1:
zero = Constant([0] * V.element().value_dimension(0))
else:
cpp.dolfin_error("bcs.py",
"homogenize boundary condition",
"Unhandled value rank %d for homogenization of boundary conditions" % \
V.element().value_rank())
# Create homogeneous boundary condition
if len(bc.domain_args) == 1:
new_bc = cpp.DirichletBC(V, zero, bc.domain_args[0])
elif len(bc.domain_args) == 2:
new_bc = cpp.DirichletBC(V, zero, bc.domain_args[0], bc.domain_args[1])
else:
cpp.dolfin_error("bcs.py", "homogenize boundary condition",
"Unknown type of boundary specification")
new_bc.domain_args = bc.domain_args
return new_bc
def __init__(self, rhs_form, solution, t, bcs, a, b, c, order, \
generator=_butcher_scheme_generator):
bcs = bcs or []
t = t or Constant(0.0)
ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
k, dt, human_form, contraction = generator(\
a, b, c, t, solution, rhs_form)
# Store data
self._rhs_form = rhs_form
self._ufl_stage_forms = ufl_stage_forms
self._dolfin_stage_forms = dolfin_stage_forms
self._t = t
self._bcs = bcs
self._dt = dt
self._last_stage = last_stage
self._solution = solution
self._k = k
self.a = a
self.b = b
self.c = c
self._order = order
self.contraction = contraction
self.jacobian_indices = jacobian_indices
cpp.MultiStageScheme.__init__(self, dolfin_stage_forms, last_stage, k, \
solution, t, dt, c, jacobian_indices, order,\
self.__class__.__name__,
human_form, bcs)
def __init__(self, *args, **kwargs):
"Create Dirichlet boundary condition"
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.DirichletBC):
cpp.dolfin_error("bcs.py",
"create DirichletBC",
"Expecting a DirichleBC as only argument"\
" for copy constructor")
# Initialize base class
cpp.DirichletBC.__init__(self, args[0])
self.domain_args = args[0].domain_args
return
# Special case for value specified as float, tuple or similar
if len(args) >= 2 and not isinstance(args[1], cpp.GenericFunction):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0])
else:
expr = Constant(args[1]) # let Constant handle all problems
args = args[:1] + (expr, ) + args[2:]
# Special case for sub domain specified as a function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
sub_domain = AutoSubDomain(args[2])
args = args[:2] + (sub_domain, ) + args[3:]
# Special case for sub domain specified as a string
if len(args) >= 3 and isinstance(args[2], str):
sub_domain = CompiledSubDomain(args[2])
args = args[:2] + (sub_domain, ) + args[3:]
# Store Expression to avoid scoping issue with SWIG directors
if isinstance(args[1], cpp.Expression):
self.function_arg = args[1]
# Store SubDomain to avoid scoping issue with SWIG directors
self.domain_args = args[2:]
# FIXME: Handling of multiple default arguments does not
# really work between Python and C++ when C++ requires them to
# be ordered and cannot accept the latter without the
# former...
# Add keyword arguments (in correct order...)
allowed_kwargs = ["method", "check_midpoint"]
for key in allowed_kwargs:
if key in kwargs:
args = tuple(list(args) + [kwargs[key]])
# Check for other keyword arguments
for key in kwargs:
if not key in allowed_kwargs:
cpp.dolfin_error("bcs.py", "create boundary condition",
"Unknown keyword argument \"%s\"" % key)
# Initialize base class
cpp.DirichletBC.__init__(self, *args)
def __init__(self,
rhs_form,
solution,
time,
bcs,
a,
b,
c,
order,
generator=_butcher_scheme_generator):
bcs = bcs or []
time = time or Constant(0.0)
ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
stage_solutions, dt, human_form, contraction = \
generator(a, b, c, time, solution, rhs_form)
# Store data
self.a = a
self.b = b
self.c = c
MultiStageScheme.__init__(self, rhs_form, ufl_stage_forms,
dolfin_stage_forms, last_stage,
stage_solutions, solution, time, dt,
c, jacobian_indices, order,\
self.__class__.__name__, human_form,
bcs, contraction)
def __init__(self,
rhs_form,
solution,
time,
order,
generalized,
generator=_rush_larsen_scheme_generator):
# FIXME: What with bcs?
bcs = []
time = time or Constant(0.0)
if order not in [1, 2]:
raise ValueError("Expected order to be either 1 or 2")
rhs_form, ufl_stage_forms, linear_terms, dofin_stage_forms, last_stage, \
stage_solutions, dt, dt_stage_offsets, human_form, contraction = \
generator(rhs_form, solution, time, order, generalized)
self.linear_terms = linear_terms
# All stages are explicit
jacobian_indices = [-1 for _ in stage_solutions]
# Highjack a and b to hold parameters for RushLarsen scheme generation
MultiStageScheme.__init__(self, rhs_form, ufl_stage_forms,
dofin_stage_forms, last_stage,
stage_solutions, solution, time, dt,
dt_stage_offsets, jacobian_indices, order,
self.__class__.__name__, human_form, bcs,
contraction)
def _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms, system_size, \
y0, stage_solution, dt, time, a, c, v, DX, time_dep_expressions):
# If we need to replace the original solution with stage solution
repl = None
if stage_solution is not None:
if system_size > 1:
repl = {y0: stage_solution}
else:
repl = {y0[0]: stage_solution}
# If we have time dependent expressions
if time_dep_expressions and abs(float(c)) > DOLFIN_EPS:
time_ = time
time = time + dt * float(c)
repl.update(_replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, float(c)))
repl[time_] = time
# If all terms are linear (using generalized=True) we add a safe
# guard to the linearized term. See below
safe_guard = sum(linear_terms) == system_size
# Add componentwise contribution to rl form
rl_ufl_form = ufl.zero()
num_rl_steps = 0
for ind in range(system_size):
# forward euler step
fe_du_i = rhs_exprs[ind] * dt * float(a)
# If exact integration
if linear_terms[ind]:
num_rl_steps += 1
# Rush Larsen step
# Safeguard the divisor: let's hope diff_rhs_exprs[ind] is never 1.0e-16!
# Let's get rid of this when the conditional fixes land properly in UFL.
eps = Constant(1.0e-16)
rl_du_i = rhs_exprs[ind]/(diff_rhs_exprs[ind] + eps)*(\
ufl.exp(diff_rhs_exprs[ind]*dt) - 1.0)
# If safe guard
if safe_guard:
du_i = ufl.conditional(ufl.lt(abs(diff_rhs_exprs[ind]), 1e-8),
fe_du_i, rl_du_i)
else:
du_i = rl_du_i
else:
du_i = fe_du_i
# If we should replace solution in form with stage solution
if repl:
du_i = ufl.replace(du_i, repl)
rl_ufl_form += (y0[ind] + du_i) * v[ind]
return rl_ufl_form * DX
def _find_linear_terms(rhs_exprs, u):
"""
Help function that takes a list of rhs expressions and return a
list of bools determining what component, rhs_exprs[i], is linear
wrt u[i].
"""
uu = [Constant(1.0) for _ in rhs_exprs]
if len(rhs_exprs) > 1:
repl = {u: ufl.as_vector(uu)}
else:
repl = {u: uu[0]}
linear_terms = []
for i, ui in enumerate(uu):
comp_i_s = expand_indices(ufl.replace(rhs_exprs[i], repl))
linear_terms.append(ui in extract_coefficients(comp_i_s) and \
ui not in extract_coefficients(\
expand_derivatives(ufl.diff(comp_i_s, ui))))
return linear_terms
def __init__(self, rhs_form, solution, t, bcs, a, b, c, order):
bcs = bcs or []
t = t or Constant(0.0)
ufl_stage_forms, dolfin_stage_forms, last_stage, k, dt, human_form = \
_butcher_scheme_generator(a, b, c, solution, rhs_form)
# Store data
self._rhs_form = rhs_form
self._ufl_stage_forms = ufl_stage_forms
self._dolfin_stage_forms = dolfin_stage_forms
self._t = t
self._dt = dt
self._last_stage = last_stage
self._solution = solution
self._k = k
self.a = a
self.b = b
self.c = c
cpp.MultiStageScheme.__init__(self, dolfin_stage_forms, last_stage, k, \
solution, t, dt, c, order,
self.__class__.__name__,
human_form, bcs)
def _butcher_scheme_generator_adm(a, b, c, time, solution, rhs_form, adj):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
adj (_Function_)
The derivative of the functional with respect to y_n+1
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kbar = [Function(solution.function_space(), name="kbar_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
# The recomputation of the forward run:
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(\
time_dep_expressions, time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [stage_form_implicit, derivative(\
stage_form_implicit, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
for i, kbari in reversed(list(enumerate(kbar))):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# And now the adjoint linearisation:
stage_form_adm = ufl.inner(dt * b[i] * adj, v)*DX + sum(\
[dt * float(a[j,i]) * safe_action(safe_adjoint(derivative(\
forward_forms[j], y_)), kbar[j]) for j in range(i, size)])
if explicit:
stage_forms_adm = [stage_form_adm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_adm -= ufl.inner(kbar[i], v) * DX
stage_forms_adm = [
stage_form_adm,
derivative(stage_form_adm, kbari)
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_adm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_adm])
stage_solutions.append(kbari)
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(adj, v)*DX + sum(\
[safe_action(safe_adjoint(derivative(forward_forms[i], y_)), kbar[i]) \
for i in range(size)]))
else:
raise Exception("Not sure what to do here")
human_form = "unimplemented"
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage,\
stage_solutions, dt, human_form, adj
def _butcher_scheme_generator_tlm(a, b, c, time, solution, rhs_form,
perturbation):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
perturbation (_Function_)
The perturbation in the initial condition of the solution
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
kdot = [Function(solution.function_space(), name="kdot_%d"%i) \
for i in range(size)]
# Create the stage forms
y_ = solution
ydot = solution.copy()
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
forward_forms = []
stage_solutions = []
jacobian_indices = []
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
forward_forms.append(stage_form)
# The recomputation of the forward run:
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_implicit = stage_form - ufl.inner(ki, v) * DX
stage_forms = [
stage_form_implicit,
derivative(stage_form_implicit, ki)
]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
stage_solutions.append(ki)
# And now the tangent linearisation:
stage_form_tlm = safe_action(derivative(stage_form, y_), perturbation) + \
sum([dt*float(a[i,j]) * safe_action(derivative(\
forward_forms[j], y_), kdot[j]) for j in range(i+1)])
if explicit:
stage_forms_tlm = [stage_form_tlm]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form_tlm -= ufl.inner(kdot[i], v) * DX
stage_forms_tlm = [
stage_form_tlm,
derivative(stage_form_tlm, kdot[i])
]
jacobian_indices.append(1)
ufl_stage_forms.append(stage_forms_tlm)
dolfin_stage_forms.append([Form(form) for form in stage_forms_tlm])
stage_solutions.append(kdot[i])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(perturbation + sum(\
[dt*float(bi)*kdoti for bi, kdoti in zip(b, kdot)], zero_), v)*DX)
else:
raise Exception("Not sure what to do here")
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
kdotterm = " + ".join("%(a)sh*action(derivative(f(t_n%(cih)s, y_n + "\
"%(kterm)s), kdot_%(i)s" % \
{"a": ("" if a[i,j] == 1.0 else "%s*"% a[i,j], j),
"i": i,
"cih": cih,
"kterm": kterm} \
for j in range(size) if a[i,j] != 0)
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
human_form.append("kdot_%(i)s = action(derivative("\
"f(t_n%(cih)s, y_n), y_n), ydot_n)" % \
{"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
human_form.append("kdot_%(i)s = action(derivative(f(t_n%(cih)s, "\
"y_n + %(kterm)s), y_n) + %(kdotterm)s" % \
{"i": i, "cih": cih, "kterm": kterm, "kdotterm": kdotterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("ydot_{n+1} = ydot_n + h*" + parentheses % (" + ".join(\
"%skdot_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
stage_solutions, dt, human_form, perturbation
def _butcher_scheme_generator(a, b, c, time, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
time (_Constant_)
A Constant holding the time at the start of the time step
solution (_Function_)
The prognostic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
a = _check_abc(a, b, c)
size = a.shape[0]
DX = _check_form(rhs_form)
# Get test function
arguments, coefficients = ufl.algorithms.\
extract_arguments_and_coefficients(rhs_form)
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [
Function(solution.function_space(), name="k_%d" % i)
for i in range(size)
]
jacobian_indices = []
# Create the stage forms
y_ = solution
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.shape())
for i, ki in enumerate(k):
# Check whether the stage is explicit
explicit = a[i, i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], zero_)
time = time_ + dt * c[i]
replace_dict = _replace_dict_time_dependent_expression(time_dep_expressions, \
time_, dt, c[i])
replace_dict[y_] = evalargs
replace_dict[time_] = time
stage_form = ufl.replace(rhs_form, replace_dict)
if explicit:
stage_forms = [stage_form]
jacobian_indices.append(-1)
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v) * DX
stage_forms = [stage_form, derivative(stage_form, ki)]
jacobian_indices.append(0)
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b, k)], zero_), v)*DX)
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[0,:], k)], zero_), v)*DX),
Form(ufl.inner(y_+sum([dt*float(bi)*ki for bi, ki in \
zip(b[1,:], k)], zero_), v)*DX)]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {
"i": i,
"cih": cih
})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b > 0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, jacobian_indices, last_stage, \
k, dt, human_form, None
def _butcher_scheme_generator(a, b, c, solution, rhs_form):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
a (2 dimensional numpy array)
The a matrix of the Butcher tableau.
b (1-2 dimensional numpy array)
The b vector of the Butcher tableau. If b is 2 dimensional the
scheme includes an error estimator and can be used in adaptive
solvers.
c (1 dimensional numpy array)
The c vector the Butcher tableau.
solution (_Function_)
The prognastic variable
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
"""
if not (isinstance(a, np.ndarray) and (len(a) == 1 or \
(len(a.shape)==2 and a.shape[0] == a.shape[1]))):
raise TypeError("Expected an m x m numpy array as the first argument")
if not (isinstance(b, np.ndarray) and len(b.shape) in [1,2]):
raise TypeError("Expected a 1 or 2 dimensional numpy array as the second argument")
if not (isinstance(c, np.ndarray) and len(c.shape) == 1):
raise TypeError("Expected a 1 dimensional numpy array as the third argument")
# Make sure a is a "matrix"
if len(a) == 1:
a.shape = (1, 1)
# Get size of system
size = a.shape[0]
# If b is a matrix we expect it to have two rows
if len(b.shape) == 2:
if not (b.shape[0] == 2 and b.shape[1] == size):
raise ValueError("Expected a 2 row matrix with the same number "\
"of collumns as the first dimension of the a matrix.")
elif len(b) != size:
raise ValueError("Expected the length of the b vector to have the "\
"same size as the first dimension of the a matrix.")
if len(c) != size:
raise ValueError("Expected the length of the c vector to have the "\
"same size as the first dimension of the a matrix.")
# Check if tableau is fully implicit
for i in range(size):
for j in range(i):
if a[j, i] != 0:
raise ValueError("Does not support fully implicit Butcher tableau.")
if not isinstance(rhs_form, ufl.Form):
raise TypeError("Expected a ufl.Form as the 5th argument.")
# Check if form contains a cell or point integral
if "cell" in rhs_form.integral_groups():
DX = ufl.dx
elif "point" in rhs_form.integral_groups():
DX = ufl.dP
else:
raise ValueError("Expected either a cell or point integral in the form.")
# Get test function
arguments, coefficients = ufl.algorithms.extract_arguments_and_coefficients(rhs_form)
if len(arguments) != 1:
raise ValueError("Expected the form to have rank 1")
v = arguments[0]
# Create time step
dt = Constant(0.1)
# rhs forms
dolfin_stage_forms = []
ufl_stage_forms = []
# Stage solutions
k = [solution.copy(deepcopy=True) for i in range(size)]
# Create the stage forms
y_ = solution
for i, ki in enumerate(k):
# Check wether the stage is explicit
explicit = a[i,i] == 0
# Evaluation arguments for the ith stage
evalargs = y_ + dt * sum([float(a[i,j]) * k[j] \
for j in range(i+1)], ufl.zero(*y_.shape()))
stage_form = ufl.replace(rhs_form, {y_:evalargs})
if explicit:
stage_forms = [stage_form]
else:
# Create a F=0 form and differentiate it
stage_form -= ufl.inner(ki, v)*DX
stage_forms = [stage_form, derivative(stage_form, ki)]
ufl_stage_forms.append(stage_forms)
dolfin_stage_forms.append([Form(form) for form in stage_forms])
# Only one last stage
if len(b.shape) == 1:
last_stage = cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b, k)])
else:
# FIXME: Add support for addaptivity in RKSolver and MultiStageScheme
last_stage = [cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[0,:], k)]),
cpp.FunctionAXPY([(float(bi), ki) for bi, ki in zip(b[1,:], k)])]
# Create the Function holding the solution at end of time step
#k.append(solution.copy())
# Generate human form of MultiStageScheme
human_form = []
for i in range(size):
kterm = " + ".join("%sh*k_%s" % ("" if a[i,j] == 1.0 else \
"%s*"% a[i,j], j) \
for j in range(size) if a[i,j] != 0)
if c[i] in [0.0, 1.0]:
cih = " + h" if c[i] == 1.0 else ""
else:
cih = " + %s*h" % c[i]
if len(kterm) == 0:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n)" % {"i": i, "cih": cih})
else:
human_form.append("k_%(i)s = f(t_n%(cih)s, y_n + %(kterm)s)" % \
{"i": i, "cih": cih, "kterm": kterm})
parentheses = "(%s)" if np.sum(b>0) > 1 else "%s"
human_form.append("y_{n+1} = y_n + h*" + parentheses % (" + ".join(\
"%sk_%s" % ("" if b[i] == 1.0 else "%s*" % b[i], i) \
for i in range(size) if b[i] > 0)))
human_form = "\n".join(human_form)
return ufl_stage_forms, dolfin_stage_forms, last_stage, k, dt, human_form
def _rush_larsen_scheme_generator(rhs_form, solution, time, order,
generalized):
"""
Generates a list of forms and solutions for a given Butcher tableau
*Arguments*
rhs_form (ufl.Form)
A UFL form representing the rhs for a time differentiated equation
solution (_Function_)
The prognostic variable
time (_Constant_)
A Constant holding the time at the start of the time step
order (int)
The order of the scheme
generalized (bool)
If True generate a generalized Rush Larsen scheme, linearizing all
components.
"""
DX = _check_form(rhs_form)
if DX != ufl.dP:
raise TypeError("Expected a form with a Pointintegral.")
# Create time step
dt = Constant(0.1)
# Get test function
arguments = rhs_form.arguments()
coefficients = rhs_form.coefficients()
# Get time dependent expressions
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
# Extract rhs expressions from form
rhs_integrand = rhs_form.integrals()[0].integrand()
rhs_exprs, v = extract_tested_expressions(rhs_integrand)
vector_rhs = len(v.ufl_shape) > 0 and v.ufl_shape[0] > 1
system_size = v.ufl_shape[0] if vector_rhs else 1
# Fix for indexing of v for scalar expressions
v = v if vector_rhs else [v]
# Extract linear terms if not using generalized Rush Larsen
if not generalized:
linear_terms = _find_linear_terms(rhs_exprs, solution)
else:
linear_terms = [True for _ in range(system_size)]
# Wrap the rhs expressions into a ufl vector type
rhs_exprs = ufl.as_vector([rhs_exprs[i] for i in range(system_size)])
rhs_jac = ufl.diff(rhs_exprs, solution)
# Takes time!
if vector_rhs:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[ind, ind]))\
for ind in range(system_size)]
else:
diff_rhs_exprs = [expand_indices(expand_derivatives(rhs_jac[0]))]
solution = [solution]
ufl_stage_forms = []
dolfin_stage_forms = []
dt_stage_offsets = []
# Stage solutions (3 per order rhs, linearized, and final step)
# If 2nd order the final step for 1 step is a stage
if order == 1:
stage_solutions = []
rl_ufl_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms, \
system_size, solution, None, dt, time, 1.0, \
0.0, v, DX, time_dep_expressions)
elif order == 2:
# Stage solution for order 2
if vector_rhs:
stage_solutions = [
Function(solution.function_space(), name="y_1/2")
]
else:
stage_solutions = [
Function(solution[0].function_space(), name="y_1/2")
]
stage_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms, \
system_size, solution, None, dt, time, 0.5, \
0.0, v, DX, time_dep_expressions)
rl_ufl_form = _rush_larsen_step(rhs_exprs, diff_rhs_exprs, linear_terms, \
system_size, solution, stage_solutions[0], dt,\
time, 1.0, 0.5, v, DX, time_dep_expressions)
ufl_stage_forms.append([stage_form])
dolfin_stage_forms.append([Form(stage_form)])
# Get last stage form
last_stage = Form(rl_ufl_form)
human_form = "%srush larsen %s" % ("generalized " if generalized else "",
str(order))
return rhs_form, linear_terms, ufl_stage_forms, dolfin_stage_forms, last_stage, \
stage_solutions, dt, dt_stage_offsets, human_form, None