/usr/lib/python2.7/dist-packages/dolfin/multistage/multistagescheme.py is in python-dolfin 1.3.0+dfsg-2.
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 | """This module defines different MultiStageScheme classes which can be passed to a RKSolver"""
# Copyright (C) 2013 Johan Hake
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
#
# First added: 2013-02-22
# Last changed: 2013-02-22
import numpy as np
# Import SWIG-generated extension module (DOLFIN C++)
import dolfin.cpp as cpp
# Import ufl
import ufl
# Import classes from dolfin python layer
from dolfin.functions.constant import Constant
from dolfin.fem.formmanipulations import derivative
from dolfin.fem.form import Form
# FIXME: Add support for algebraic parts (at least for implicit)
# FIXME: Add support for implicit/explicit split ala IMEX schemes
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
class MultiStageScheme(cpp.MultiStageScheme):
"""
Base class for all MultiStageSchemes
"""
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 rhs_form(self):
"Return the original rhs form"
return self._rhs_form
def ufl_stage_forms(self):
"Return the ufl stage forms"
return self._ufl_stage_forms
def dolfin_stage_forms(self):
"Return the dolfin stage forms"
return self._dolfin_stage_forms
def t(self):
"Return the Constant used to describe time in the MultiStageScheme"
return self._t
def dt(self):
"Return the Constant used to describe time in the MultiStageScheme"
return self._dt
def solution(self):
"Return the solution Function"
return self._solution
def last_stage(self):
"Return the AXPYFunction object describing the last stage"
return self._last_stage
def stage_solutions(self):
"Return the stage solutions"
return self._stage_solutions
class ERK1(MultiStageScheme):
"""
Explicit first order Scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([0.])
b = np.array([1.])
c = np.array([0.])
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 1)
class BDF1(MultiStageScheme):
"""
Implicit first order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([1.])
b = np.array([1.])
c = np.array([1.])
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 1)
class ExplicitMidPoint(MultiStageScheme):
"""
Explicit 2nd order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0, 0],[0.5, 0.0]])
b = np.array([0., 1])
c = np.array([0, 0.5])
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 2)
class CN2(MultiStageScheme):
"""
Semi-implicit 2nd order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0, 0],[0.5, 0.5]])
b = np.array([0.5, 0.5])
c = np.array([0, 1.0])
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 2)
class ERK4(MultiStageScheme):
"""
Explicit 4th order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0, 0, 0, 0],
[0.5, 0, 0, 0],
[0, 0.5, 0, 0],
[0, 0, 1, 0]])
b = np.array([1./6, 1./3, 1./3, 1./6])
c = np.array([0, 0.5, 0.5, 1])
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 4)
class ESDIRK3(MultiStageScheme):
"""
Explicit implicit 3rd order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[ 0. , 0. , 0. , 0. , 0. ],
[ 0.43586652, 0.43586652, 0. , 0. , 0. ],
[ 0.14073777, -0.10836555, 0.43586652, 0. , 0. ],
[ 0.1023994 , -0.37687845, 0.83861253, 0.43586652, 0. ],
[ 0.1570249 , 0.11733044, 0.61667803, -0.32689989, 0.43586652]])
b = a[-1,:].copy()
c = a.sum(1)
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 3)
class ESDIRK4(MultiStageScheme):
"""
Explicit implicit 4rd order scheme
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0, 0, 0, 0, 0 ],
[0.435866521500000, 0.435866521500000, 0, 0, 0 ],
[0.140737774731968, -0.108365551378832, 0.435866521500000, 0, 0 ],
[0.102399400616089, -0.376878452267324, 0.838612530151233, 0.435866521500000, 0 ],
[0.157024897860995, 0.117330441357768, 0.616678030391680, -0.326899891110444, 0.435866521500000 ]])
b = a[-1,:].copy()
c = a.sum(1)
MultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 4)
# Aliases
CrankNicolson = CN2
ExplicitEuler = ERK1
ForwardEuler = ERK1
ImplicitEuler = BDF1
BackwardEuler = BDF1
ERK = ERK1
RK4 = ERK4
__all__ = [name for name, attr in globals().items() \
if isinstance(attr, type) and issubclass(attr, MultiStageScheme)]
__all__.append("MultiStageScheme")
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