# -*- coding: utf-8 -*-
"""This module defines different MultiStageScheme classes which can be
passed to a RKSolver or PointIntegralSolver
"""
# Copyright (C) 2013-2015 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/>.
#
# Modified by Patrick Farrell, 2013
# Modified by Martin Sandve Alnæs, 2015
import numpy as np
import functools
import ufl
import dolfin.cpp as cpp
from dolfin.function.constant import Constant
from dolfin.function.expression import Expression
from dolfin.function.function import Function
from dolfin.fem.formmanipulations import derivative, adjoint
from ufl import action as ufl_action
from dolfin.fem.form import Form
import ufl.algorithms
from ufl.algorithms import expand_derivatives
# FIXME: Add support for algebraic parts (at least for implicit)
# FIXME: Add support for implicit/explicit split ala IMEX schemes
def safe_adjoint(x):
return adjoint(x, reordered_arguments=x.arguments())
def safe_action(x, y):
x = expand_derivatives(x)
if x.integrals() == ():
return x # form is empty, return anyway
else:
return ufl_action(x, y)
def _check_abc(a, b, c):
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 the method is singly diagonally implicit
sigma = -1
for i in range(size):
# If implicit
if a[i, i] != 0:
if sigma == -1:
sigma = a[i, i]
elif sigma != a[i, i]:
raise ValueError("Expected only singly diagonally implicit "
"schemes. (Same value on the diagonal of 'a'.)")
# 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.")
return a
def _check_form(rhs_form):
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 rhs_form.integrals_by_type("cell"):
DX = ufl.dx
elif rhs_form.integrals_by_type("vertex"):
DX = ufl.dP
else:
raise ValueError("Expected either a cell or vertex integral in the form.")
if len(rhs_form.integrals()) != 1:
raise ValueError("Expected only one integral in form.")
arguments = rhs_form.arguments()
if len(arguments) != 1:
raise ValueError("Expected the form to have rank 1")
return DX
def _time_dependent_expressions(rhs_form, time):
"""Return a list of expressions which uses the present time as a
parameter
"""
# FIXME: Add extraction of time dependant expressions from bcs too
time_dependent_expressions = dict()
for coefficient in rhs_form.coefficients():
if hasattr(coefficient, "_user_parameters"):
for c_name, c in list(coefficient._user_parameters.items()):
if isinstance(c, ufl.Coefficient) and time.id() == c.id():
if coefficient not in time_dependent_expressions:
time_dependent_expressions[coefficient] = [c_name]
else:
time_dependent_expressions[coefficient].append(c_name)
return time_dependent_expressions
def _replace_dict_time_dependent_expression(time_dep_expressions, time,
dt, c):
assert(isinstance(c, float))
replace_dict = {}
if c == 0.0 or not time_dep_expressions:
return replace_dict
new_time = Expression("time + c*dt", time=time, c=c, dt=dt, degree=0)
for expr, c_names in list(time_dep_expressions.items()):
assert(isinstance(expr, Expression))
kwargs = dict(name=expr.name(), label=expr.label(),
element=expr.ufl_element(), **expr._user_parameters)
for c_name in c_names:
kwargs[c_name] = new_time
replace_dict[expr] = Expression(expr._cppcode, **kwargs)
return replace_dict
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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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_.ufl_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 adaptivity 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_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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_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_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 = rhs_form.arguments()
# coefficients = rhs_form.coefficients()
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
time_ = time
time_dep_expressions = _time_dependent_expressions(rhs_form, time)
zero_ = ufl.zero(*y_.ufl_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
[docs]class MultiStageScheme(cpp.multistage.MultiStageScheme):
"""Base class for all MultiStageSchemes
"""
def __init__(self, rhs_form, ufl_stage_forms,
dolfin_stage_forms, last_stage, stage_solutions,
solution, time, dt, dt_stage_offsets, jacobian_indices, order,
name, human_form, bcs, contraction=None):
# Store Python data
self._rhs_form = rhs_form
self._ufl_stage_forms = ufl_stage_forms
self._dolfin_stage_forms = dolfin_stage_forms
self._t = time
self._bcs = bcs
self._dt = dt
self._last_stage = last_stage
self._solution = solution
self._stage_solutions = stage_solutions
self._order = order
self.jacobian_indices = jacobian_indices
self.contraction = contraction
# Pass args to C++ constructor
stage_solutions = [s.cpp_object() for s in stage_solutions]
cpp.multistage.MultiStageScheme.__init__(self,
dolfin_stage_forms,
last_stage,
stage_solutions,
solution.cpp_object(),
time.cpp_object(),
dt.cpp_object(),
dt_stage_offsets,
jacobian_indices,
order,
self.__class__.__name__,
human_form, bcs)
[docs] def t(self):
"Return the Constant used to describe time in the MultiStageScheme"
return self._t
[docs] def dt(self):
"Return the Constant used to describe time in the MultiStageScheme"
return self._dt
[docs] def solution(self):
"Return the solution Function"
return self._solution
[docs] def last_stage(self):
"Return the form describing the last stage"
return self._last_stage
[docs] def stage_solutions(self):
"Return the stage solutions"
return self._stage_solutions
def to_tlm(self, perturbation):
raise NotImplementedError("'to_tlm:' implement in derived classes")
def to_adm(self, perturbation):
raise NotImplementedError("'to_adm:' implement in derived classes")
[docs]class ButcherMultiStageScheme(MultiStageScheme):
"""Base class for all MultiStageSchemes
"""
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)
[docs] def to_tlm(self, perturbation):
r"""Return another MultiStageScheme that implements the tangent
linearisation of the ODE solver.
This takes \dot{y_n} (the derivative of y_n with respect to a
parameter) and computes \dot{y_n+1} (the derivative of y_n+1
with respect to that parameter).
"""
generator = functools.partial(_butcher_scheme_generator_tlm,
perturbation=perturbation)
new_solution = self._solution.copy()
new_form = ufl.replace(self._rhs_form, {self._solution: new_solution})
return ButcherMultiStageScheme(new_form, new_solution,
self._t, self._bcs, self.a,
self.b, self.c, self._order,
generator=generator)
[docs] def to_adm(self, adj):
r"""Return another MultiStageScheme that implements the adjoint
linearisation of the ODE solver.
This takes \bar{y_n+1} (the derivative of a functional J with
respect to y_n+1) and computes \bar{y_n} (the derivative of J
with respect to y_n).
"""
generator = functools.partial(_butcher_scheme_generator_adm, adj=adj)
new_solution = self._solution.copy()
new_form = ufl.replace(self._rhs_form, {self._solution: new_solution})
return ButcherMultiStageScheme(new_form, new_solution, self._t, self._bcs,
self.a, self.b, self.c, self._order,
generator=generator)
[docs]class ERK1(ButcherMultiStageScheme):
"""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.])
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs,
a, b, c, 1)
[docs]class BDF1(ButcherMultiStageScheme):
"""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.])
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs,
a, b, c, 1)
[docs]class ExplicitMidPoint(ButcherMultiStageScheme):
"""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])
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs,
a, b, c, 2)
[docs]class CN2(ButcherMultiStageScheme):
"""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])
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs,
a, b, c, 2)
[docs]class ERK4(ButcherMultiStageScheme):
"""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])
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs,
a, b, c, 4)
[docs]class ESDIRK3(ButcherMultiStageScheme):
"""Explicit implicit 3rd order scheme
See also "Singly diagonally implicit Runge–Kutta methods with an
explicit first stage" by A Kværnø - BIT Numerical Mathematics,
2004 (p.497)
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0.000000000000000, 0.000000000000000, 0.000000000000000, 0.00000000000000],
[0.435866521500000, 0.435866521500000, 0.000000000000000, 0.00000000000000],
[0.490563388419108, 0.073570090080892, 0.435866521500000, 0.00000000000000],
[0.308809969973036, 1.490563388254108, -1.235239879727145, 0.435866521500000]])
b = a[-1, :].copy()
c = a.sum(1)
ButcherMultiStageScheme.__init__(self, rhs_form, solution, t, bcs, a, b, c, 3)
[docs]class ESDIRK4(ButcherMultiStageScheme):
"""Explicit implicit 4rd order scheme
See also "Singly diagonally implicit Runge–Kutta methods with an
explicit first stage" by A Kværnø - BIT Numerical Mathematics,
2004 (p.498)
"""
def __init__(self, rhs_form, solution, t=None, bcs=None):
a = np.array([[0.000000000000000, 0.0000000000000000, 0.000000000000000, 0.000000000000000, 0.000000000000000],
[0.435866521500000, 0.4358665215000000, 0.000000000000000, 0.000000000000000, 0.000000000000000],
[0.140737774731968, -0.108365551378832, 0.435866521500000, 0.000000000000000, 0.000000000000000],
[0.102399400616089, -0.376878452267324, 0.838612530151233, 0.435866521500000, 0.000000000000000],
[0.157024897860995, 0.1173304413577680, 0.616678030391680, -0.326899891110444, 0.435866521500000]])
b = a[-1, :].copy()
c = a.sum(1)
ButcherMultiStageScheme.__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 list(globals().items())
if isinstance(attr, type) and issubclass(attr, MultiStageScheme)]
__all__.append("MultiStageScheme")