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Exponential Integrators and Lie groupmethods
Workshop on Exponential Integrators
October 20-23, 2004, Innsbruck.
Borislav V. [email protected]
http://www.ii.uib.no/ �borko
Department of computer science
University of Bergen, Norway
Exponential Integrators and Lie group methods – p.1/21
Outline
�
The framework
�
The choice of action
�
LGI for semilinear problems
�
Implementation issues
�
Numerical experiments
�
Conclusions
�
Future work
Exponential Integrators and Lie group methods – p.2/21
Exp. int. and Lie group methods
Lie group methods
�
Designed to preserve certain qualitatively properties of the exact flow
The freedom in the choice of the action allows to define the basic motions on themanifold in such a way that they provide a good approximation to the flow of theoriginal vector field.
Exponential Integrators and Lie group methods – p.3/21
Exp. int. and Lie group methods
Lie group methods
�
Designed to preserve certain qualitatively properties of the exact flow�
The freedom in the choice of the action allows to define the basic motions on themanifold in such a way that they provide a good approximation to the flow of theoriginal vector field.
Exponential Integrators and Lie group methods – p.3/21
Background theory
Lie group -
�� � � � � �
Lie algebra � �� � - linear space with bracket
� � � � � � � ���� ��
��������� � � � � � � � � � � � � � � � � � � � � �� � �
where � � � � �
and � � � � �
are smooth curves in
�
such that � � � � � � � � � and� ! � � � � � � � ! � � � � � .
Define the product by
where is a smooth curve in such that and .
Exponential Integrators and Lie group methods – p.4/21
Background theory
Lie group -
�� � � � � �
Lie algebra � �� � - linear space with bracket
� � � � � � � ���� ��
��������� � � � � � � � � � � � � � � � � � � � � �� � �
where � � � � �
and � � � � �
are smooth curves in
�
such that � � � � � � � � � and� ! � � � � � � � ! � � � � � .Define the product "$# � % � & �
by
� "(' ))�
�*���*�� � � � � � � �' �
where � � � �
is a smooth curve in�
such that � � � � and � ! � � � .
Exponential Integrators and Lie group methods – p.4/21
The Exp map
The exponential map provides connection between
�
and �.Exp# � & � is defined as Exp
� � � � �+ �
, where � � � �$, �
satisfies the differentialequation � � � � ! � " � � � � � � � � �.-
The differential of the exponential map dExp is defined as the right trivializedtangent
dExp Exp Exp
dExpe
adad
dExpe ad
ad
where the coefficients are the Bernoulli numbers and
ad ad ad for
Exponential Integrators and Lie group methods – p.5/21
The Exp map
The exponential map provides connection between
�
and �.Exp# � & � is defined as Exp
� � � � �+ �
, where � � � �$, �
satisfies the differentialequation � � � � ! � " � � � � � � � � �.-The differential of the exponential map dExp# � % � & � is defined as the right trivializedtangent
dExp
� /� � � �
Exp
� /� � ))�
������� � � � Exp
� /� 0 � � � -
dExpe
adad
dExpe ad
ad
where the coefficients are the Bernoulli numbers and
ad ad ad for
Exponential Integrators and Lie group methods – p.5/21
The Exp map
The exponential map provides connection between
�
and �.Exp# � & � is defined as Exp
� � � � �+ �
, where � � � �$, �
satisfies the differentialequation � � � � ! � " � � � � � � � � �.-The differential of the exponential map dExp# � % � & � is defined as the right trivializedtangent
dExp
� /� � � �
Exp
� /� � ))�
������� � � � Exp
� /� 0 � � � -
dExp 1�2 � � � e
354 +6
�*����� 3 �ad 7$8 9 2 : ;<� �
+�= 0 + �> ad
<1�2 � � � �
dExp
� �1�2 � � � 6e 3?4 +
������� 3 �ad 7(8 9 2 : ;<� �
@ <= > ad
<1�2 � � � �
where the coefficients@ <are the Bernoulli numbers and
ad<1A2 � � � ad 1A2 �ad
<� �1 2 � � /� � - - - � /� � � � � � � for
= B + -
Exponential Integrators and Lie group methods – p.5/21
Actions on the manifold
A group action on a manifold
C
is a smooth map D # � % C & Csatisfying� D E E F E, C �
' D �= D E � �' � = � D E F' � = , � � E, C -
An algebra action on is given by
Exp
Note: The algebra action is not uniquely determined by the group action. Everydiffeomorphism
such that and where is the identity of the algebra, defines an
algebra action by the formula .
Exponential Integrators and Lie group methods – p.6/21
Actions on the manifold
A group action on a manifold
C
is a smooth map D # � % C & Csatisfying� D E E F E, C �
' D �= D E � �' � = � D E F' � = , � � E, C -An algebra action G # � % C & C
on
C
is given by
� G E Exp
� � � D E-
Note: The algebra action is not uniquely determined by the group action. Everydiffeomorphism
such that and where is the identity of the algebra, defines an
algebra action by the formula .
Exponential Integrators and Lie group methods – p.6/21
Actions on the manifold
A group action on a manifold
C
is a smooth map D # � % C & Csatisfying� D E E F E, C �
' D �= D E � �' � = � D E F' � = , � � E, C -An algebra action G # � % C & C
on
C
is given by
� G E Exp
� � � D E-Note: The algebra action is not uniquely determined by the group action. Everydiffeomorphism H # � & � �
such that
H � � � and
H ! � � I � where
I
is the identity of the algebra, defines an
algebra action by the formula
� G E H � � � D E.
Exponential Integrators and Lie group methods – p.6/21
Generic presentation of a diff. eq.
The product J # � % C & C
between � and
C
is defined by
� J E ))�
�*����� � � � � � � � D E �where � � � �
is a smooth curve in
�
such that � � � � and � ! � � � .
For a fix the product gives a vector field on
The vector field is called a frozen vector field.
Every differential equation evolving on a homogeneous space can always be writtenas
where .
Exponential Integrators and Lie group methods – p.7/21
Generic presentation of a diff. eq.
The product J # � % C & C
between � and
C
is defined by
� J E ))�
�*����� � � � � � � � D E �where � � � �
is a smooth curve in
�
such that � � � � and � ! � � � .
For a fix
� , � the product
� J E gives a vector field on
C
K 2 � E � � J E ))�
������� � � �LNM O � � � � D E-
The vector field
K 2 is called a frozen vector field.
Every differential equation evolving on a homogeneous space can always be writtenas
where .
Exponential Integrators and Lie group methods – p.7/21
Generic presentation of a diff. eq.
The product J # � % C & C
between � and
C
is defined by
� J E ))�
�*����� � � � � � � � D E �where � � � �
is a smooth curve in
�
such that � � � � and � ! � � � .
For a fix
� , � the product
� J E gives a vector field on
C
K 2 � E � � J E ))�
������� � � �LNM O � � � � D E-
The vector field
K 2 is called a frozen vector field.
Every differential equation evolving on a homogeneous space
C
can always be writtenas P ! � � � Q � P � J P � P � � � � P � �
where
Q# C & �.Exponential Integrators and Lie group methods – p.7/21
Lie group integrators
Algotithm. (Crouch–Grossman’93)
for
R + � - - - � � doS?T Exp
�UWV T � Q � � D D D Exp
�U V T � Q � � D PYXQT Q � S?T �
endPZX [ � Exp
�U \ � Q � � D D D Exp
�U \ � Q � � D P]X
Algotithm. (Crouch–Grossman’93)
for do
end
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for do
d
end
The values on can be computed via the formula .
Exponential Integrators and Lie group methods – p.8/21
Lie group integrators
Algotithm. (Crouch–Grossman’93)
for
R + � - - - � � doS?T �U V T � Q � � G D D D G �U V T � Q � � G PYXQT Q � S?T �
endPZX [ � �U \ � Q � � G D D D G �U \ � Q � � G P]X
Algotithm. (Crouch–Grossman’93)
for do
end
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for do
d
end
The values on can be computed via the formula .
Exponential Integrators and Lie group methods – p.8/21
Lie group integrators
Algotithm. (Crouch–Grossman’93)
for
R + � - - - � � doS?T �U V T � Q � � G D D D G �U V T � Q � � G PYXQT Q � S?T �
endPZX [ � �U \ � Q � � G D D D G �U \ � Q � � G P]X
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for
R + � - - - � � do� T U ^ �`_ � �V T _ a _QT Q � H � � T � D P]X �
a T d H� � � QT �
endPNX [ � H �U ^ �T � � \ T a T � D P]X
Algotithm. (Crouch–Grossman’93)
for do
end
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for do
d
end
The values on can be computed via the formula .
Exponential Integrators and Lie group methods – p.8/21
Lie group integrators
Algotithm. (Crouch–Grossman’93)
for
R + � - - - � � doS?T �U V T � Q � � G D D D G �U V T � Q � � G PYXQT Q � S?T �
endPZX [ � �U \ � Q � � G D D D G �U \ � Q � � G P]X
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for
R + � - - - � � do� T U ^ �`_ � �V T _ a _QT Q � � � T � G P]X �
a T d H� � � QT �
endPNX [ � �U ^ �T � � \ T a T � G P]X
The values on can be computed via the formula .
Exponential Integrators and Lie group methods – p.8/21
Lie group integrators
Algotithm. (Crouch–Grossman’93)
for
R + � - - - � � doS?T �U V T � Q � � G D D D G �U V T � Q � � G PYXQT Q � S?T �
endPZX [ � �U \ � Q � � G D D D G �U \ � Q � � G P]X
Algotithm. (Runge–Kutta Munthe-Kaas’99)
for
R + � - - - � � do� T U ^ �`_ � �V T _ a _QT Q � � � T � G P]X �
a T d H� � � QT �
endPNX [ � �U ^ �T � � \ T a T � G P]XThe values on
Ccan be computed via the formula
S T � T G P]X .
Exponential Integrators and Lie group methods – p.8/21
Lie group integrators
Algotithm. Commutator-free Lie group method (Celledoni, Marthinsen, Owren’03)
for
R + � - - - � � doS?T Exp
�U ^ � <� �V <T b Q <� D D D Exp
�U ^ � <� �V <T � Q <� D PYXQT Q � S?T �
endPZX [ � Exp
�U ^ � <� � \ <b Q <� D D D Exp
�U ^ � <� � \ <� Q <� D P]X
Algotithm. Commutator-free Lie group method (Celledoni, Marthinsen, Owren’03)
for do
end
An example of a fourth order method, based on the classical fourth order Runge–Kuttamethod
0
0
-
0
0
0
1
-
Exponential Integrators and Lie group methods – p.9/21
Lie group integrators
Algotithm. Commutator-free Lie group method (Celledoni, Marthinsen, Owren’03)
for
R + � - - - � � doS?T �U ^ � <� �V <T b Q <� G D D D G �U ^ � <� �V <T � Q <� G PYXQT Q � S?T �
endPZX [ � �U ^ � <� � \ <b Q <� G D D D G �U ^ � <� � \ <� Q <� G P]X
An example of a fourth order method, based on the classical fourth order Runge–Kuttamethod
0
0
-
0
0
0
1
-
Exponential Integrators and Lie group methods – p.9/21
Lie group integrators
Algotithm. Commutator-free Lie group method (Celledoni, Marthinsen, Owren’03)
for
R + � - - - � � doS?T �U ^ � <� �V <T b Q <� G D D D G �U ^ � <� �V <T � Q <� G PYXQT Q � S?T �
endPZX [ � �U ^ � <� � \ <b Q <� G D D D G �U ^ � <� � \ <� Q <� G P]XAn example of a fourth order method, based on the classical fourth order Runge–Kuttamethod
0� � � �� � 0
� �� �� �
� �-
� �0
0
0
1
cedf�gh�� �
� ie�i
�ie�i
-
�� ��gced
f-
Exponential Integrators and Lie group methods – p.9/21
Basic motions on
Consider the following nonautonomous problem defined on
j k
P ! l � P � � � � P � � � � P � -By adding the trivial differential equation
� ! + , we can rewrite it in the form
m ! n � m � � � � � m � � � � m � �
where
n o
p l � P � � �+
qr � m
op P�
qr -
Consider the following nonautonomous problem defined on
By adding the trivial differential equation , we can rewrite it in the form
Define:- the basic movements on to be given by the solution of a simpler diff. equation
where approximates .- the Lie algebra to be the set of all coefficients of the frozen vector fields .
- the algebra action to be the solution of (2) at time .
Exponential Integrators and Lie group methods – p.10/21
Basic motions on
Consider the following nonautonomous problem defined on
j k
P ! l � P � � � � P � � � � P � -By adding the trivial differential equation
� ! + , we can rewrite it in the form
m ! n � m � � � � � m � � � � m � -
Define:- the basic movements on
C
to be given by the solution of a simpler diff. equation
�s � m ! K 2 � m � � m � � � � m � �
where
K 2 � m � approximatesn � m � � � �
.- the Lie algebra � to be the set of all coefficients
�
of the frozen vector fields
K 2 .- the algebra action
U � G m � to be the solution of (2) at time
� � 0 U .
Exponential Integrators and Lie group methods – p.10/21
The choice of action
t The simplest case- � uv , j k [ �w u �x y � z � { � # x y � z , j k � {, jw
.- The generic function is given by
Q |}~� �� �� � l � P � � � � � � + � �x y � z � + �
.
- The frozen vector field is
K 9� �� ��� � : |}~ � � � } � �� �� �
.
- The algebra action is
U �x y � z � { � G }~� �� � }~� [ �� �� ��� [ � � �(translations).
- The commutators are given by
� � � � � � �� � �.
In this case we recover the traditional integration schemes.
When (such a representation is always possible)
- .- The generic function is given by .
- The frozen vector field is .
- The algebra action is given by ,
where denotes the matrix exponential and is the first ETD function.- The commutators are given by .
In this case we recover the affine algebra action proposed by Munthe-Kaas’99.
Exponential Integrators and Lie group methods – p.11/21
The choice of action
t The simplest case- � uv , j k [ �w u �x y � z � { � # x y � z , j k � {, jw
.- The generic function is given by
Q |}~� �� �� � l � P � � � � � � + � �x y � z � + �
.
- The frozen vector field is
K 9� �� ��� � : |}~ � � � } � �� �� �
.
- The algebra action is
U �x y � z � { � G }~� �� � }~� [ �� �� ��� [ � � �(translations).
- The commutators are given by
� � � � � � �� � �.
In this case we recover the traditional integration schemes.t When
l � P � � � � � P � � � P 0 � � P � � �
(such a representation is always possible)
- .- The generic function is given by .
- The frozen vector field is .
- The algebra action is given by ,
where denotes the matrix exponential and is the first ETD function.- The commutators are given by .
In this case we recover the affine algebra action proposed by Munthe-Kaas’99.
Exponential Integrators and Lie group methods – p.11/21
The choice of action
t The simplest case- � uv , j k [ �w u �x y � z � { � # x y � z , j k � {, jw
.- The generic function is given by
Q |}~� �� �� � l � P � � � � � � + � �x y � z � + �
.
- The frozen vector field is
K 9� �� ��� � : |}~ � � � } � �� �� �
.
- The algebra action is
U �x y � z � { � G }~� �� � }~� [ �� �� ��� [ � � �(translations).
- The commutators are given by
� � � � � � �� � �.
In this case we recover the traditional integration schemes.t When
l � P � � � � � P � � � P 0 � � P � � �
(such a representation is always possible)
- � u �� � v �$, j k [ �� k [ ��� j k [ �w � with
� o
p �
qr � v
op x
y � z{
qr �
- .- The generic function is given by .
- The frozen vector field is .
- The algebra action is given by ,
where denotes the matrix exponential and is the first ETD function.- The commutators are given by .
In this case we recover the affine algebra action proposed by Munthe-Kaas’99.
Exponential Integrators and Lie group methods – p.11/21
The choice of action
t The simplest case- � uv , j k [ �w u �x y � z � { � # x y � z , j k � {, jw
.- The generic function is given by
Q |}~� �� �� � l � P � � � � � � + � �x y � z � + �
.
- The frozen vector field is
K 9� �� ��� � : |}~ � � � } � �� �� �
.
- The algebra action is
U �x y � z � { � G }~� �� � }~� [ �� �� ��� [ � � �(translations).
- The commutators are given by
� � � � � � �� � �.
In this case we recover the traditional integration schemes.t When
l � P � � � � � P � � � P 0 � � P � � �
(such a representation is always possible)
- � u �� � v �$, j k [ �� k [ ��� j k [ �w u � � � x y � z � { � # �, j k� k � x y � z , j k � {, jw
.- The generic function is given by
Q |}~� �� �� � � � P � � � � � � � � P � � � � � � + � � � � x y � z � + �
.
- The frozen vector field isK 9� � � �� � � � : | }~ � � � } � ~ [� �� �� �
.
- The algebra action is given by
U � � � x y � z � { � G }~� �� � } � � �~� [ �� �� �W� �� � 9 �� :�� [ � � �
,
where � �� denotes the matrix exponential and
� y � z
is the first ETD
� y T z
function.- The commutators are given by
� � � � � � � � � � � 4 � � � � � � � x y � z� 4 � � x y � z� � �
.
In this case we recover the affine algebra action proposed by Munthe-Kaas’99.Exponential Integrators and Lie group methods – p.11/21
Nonautonomous frozen vector fields
t Similarly, when
l � P � � � � � P � � � P 0 � y � z � P � � � 0 � � y � z � P � � �
.
- � u �� � v �$, j k [ �� k [ ��� j k [ �w � with
� o
p � x y � z
q
r � v o
p xy � z
{q
r �
Similarly, when .
-.
- The generic function is given by.
- The frozen vector field is,
where and .
- The algebra action is given by
.- The commutators are given by
.
Can generalize this approach to the case .
- append trivial differential equations corresponding to .
Exponential Integrators and Lie group methods – p.12/21
Nonautonomous frozen vector fields
t Similarly, when
l � P � � � � � P � � � P 0 � y � z � P � � � 0 � � y � z � P � � �
.
- � u �� � v �$, j k [ �� k [ ��� j k [ �w
u � � � x y � z � x y � z � { � # �, j k� k � x y � z � x y � z , j k � {, jw.
- The generic function is given byQ ��}~� �� � � | � � P � � � � � � � y � z � P � � � � � � � y � z � P � � � � � � + � � � � x y � z � x y � z � + �
.- The frozen vector field isK 9� � � �� ��� � �� � � � : | }~ � �� � � � x y � z � x y � z � { � J }~ � � } � ~ [��� [� � �� �
,
where � � x y � z 0 � � �+ 4 { � x y � z
and � � {x y � z.
- The algebra action is given byU � � � x y � z � x y � z � { � G }~� �� � } � � �~� [ � 9� �� � [�� � �� � :� �� � 9 �� : [ � � �� �� �W� � � � 9 �� :�� [ � � �
.- The commutators are given by � � � � � � � � � � � � � � � � x y � z� 4 � � x y � z� � � � x y � z� 4 � � x y � z� 0 { � x y � z� 4 { � x y � z� � �
.
Can generalize this approach to the case .
- append trivial differential equations corresponding to .
Exponential Integrators and Lie group methods – p.12/21
Nonautonomous frozen vector fields
t Similarly, when
l � P � � � � � P � � � P 0 � y � z � P � � � 0 � � y � z � P � � �
.
- � u �� � v �$, j k [ �� k [ ��� j k [ �w
u � � � x y � z � x y � z � { � # �, j k� k � x y � z � x y � z , j k � {, jw.
- The generic function is given byQ ��}~� �� � � | � � P � � � � � � � y � z � P � � � � � � � y � z � P � � � � � � + � � � � x y � z � x y � z � + �
.- The frozen vector field isK 9� � � �� ��� � �� � � � : | }~ � �� � � � x y � z � x y � z � { � J }~ � � } � ~ [��� [� � �� �
,
where � � x y � z 0 � � �+ 4 { � x y � z
and � � {x y � z.
- The algebra action is given byU � � � x y � z � x y � z � { � G }~� �� � } � � �~� [ � 9� �� � [�� � �� � :� �� � 9 �� : [ � � �� �� �W� � � � 9 �� :�� [ � � �
.- The commutators are given by � � � � � � � � � � � � � � � � x y � z� 4 � � x y � z� � � � x y � z� 4 � � x y � z� 0 { � x y � z� 4 { � x y � z� � �
.t Can generalize this approach to the case
l � P � � � � � P � � � P 0 ^ � <� � � < � y <z � P � � �
.
- append E trivial differential equations corresponding to
� � � � � - - - � � �
.
Exponential Integrators and Lie group methods – p.12/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
A third order Crouch–Grossman method
o * � � � � � * � ¡p
0 0 0
I
¢g � y � z
0 0 � £¤ �¥
� �¦� � i � � §�� ¨ �¥ � y � z � � �¦� � i U � � � ©� �ª � y � z � � ©� �ª U � �
0 � � §� ¤ �¥
� ¢«� � £ ¨¬� �¥ � y � z � � ¢«� U � � 4 � ¢ � � ¤� § �¥ � y � z �4 � ¢ U � � � g� © � y � z � � g� © U � � � �¥q
*�����*�¡r
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
A fourth order RKMK method with exact Exp
o � � * � � � � � * � � * ¡p
I
� � � y � z � � � �¥
� � � y � z 4 � � I � � I � � � �¥
/� y ¢ z � � �¥ � � 4 /� y � z /� y ¢ z � �¥ � � /� y � z � �¥
x � �U � � x � �U � � x ¢ �U � � �i I � �¥q
��*�����*��*¡r�
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
A fourth order RKMK method with exact Exp
where /� y � z � 6� � y � z � 6� � y � z � � � 6s � � ® � 0 Is �
/� y ¢ z � 6� � y � z � � � 6�4 I � 3 4 64 II4 � 3 �
x � �U � � +¯ � y � z 4 +° � y � z /� y ¢ z � U �s � 4 +¯ � y � z � /� y ¢ z 4 s I � /� y ¢ z � � U �s � �
x � �U � � +° /� y � z 0 +¯ /� y � z � /� y ¢ z 4 s I � /� y ¢ z � U �s � �
x ¢ �U � � +° /� y � z 4 +¯ /� y � z /� y ¢ z -
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
RKMK methods with approximations to the Exp map
With appropriate choices for the diffeomorphism
Hit is possibel to show that:
- IF RK methods are RKMK methods (Krogstad’03);- GIF/RK methods are also RKMK methods.
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
A fourth order CF methodo � � � � � * � � � � � * � � * ¡p
0 0 0 0
I
� � � y � z
0 0 0 � �± �
0
� � � y � z
0 0 � �± �
� � � y � z
-
� � � y � z 0
0
0� y � z 0
0
� �± �� �± �
c²df
�g � y � z-
�� � � y � z
�i � y � z�i � y � z
�i � y � z�i � y � z -
�� � � y � z�g � y � z
� �± �� �± �
cedf
q�����*�����*��*¡r
�
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.
A fourth order CF method
o * � � * � � � � � * � ¡p
0 0 0 0
I
� � � y � z
0 0 0 � � � �¥
0� � � y � z
0 0 � � � �¥
� � � y � z | �¥ � � | � �± � 4 I�0
� y � z | �¥ � � 0 � �¥
� � � y � z 4 � ¢ � y � z | �¥ � � � ¢ � y � z � ¢ � y � z 4 �i � y � z 0 � ¢ � y � z | �¥ � � � �¥q
*��*�����*�¡r
Other possible choice is to use nonautonomous frozen vctor fields
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.t Other possible choice is to use nonautonomous frozen vctor fields� � P � � � �X 0 � ³ 9~ � � :� ³µ´� � y � z 0 � � y � z
or
where and are the values of at the end of step number and .
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
LGI for semilinear problems
Consider the semilinear problem
�+ � P ! � P 0 � � P � � � � P � � � � P � �
t Natural choice for G is the affine action.t Other possible choice is to use nonautonomous frozen vctor fields� � P � � � �X 0 � ³ 9~ � � :� ³µ´� � y � z 0 � � y � z
or � � P � � � �X 0 � �X 4 �X � �� 0 � � � � P � � �4 s �X 0 �X � �� � �
where
�X and
�X � � are the values of
�at the end of step number ¶ and ¶4 +
.
Note: In this way we again obtain GLMs.
Exponential Integrators and Lie group methods – p.13/21
Implementation issues
Consider the ETD
� y T z
functions
� y � z � 6� � 3·4 +6 � � y T [ � z � 6� � y T z � 6�4 �T ¸6 � for
R s � ° � - - - -
A straightforward implementation suffers from cancellation errors (Kassam andTrefethen).
Numerical techniques
Decomposition methods
Krylov subspace approximations
Cauchy integral approach
Exponential Integrators and Lie group methods – p.14/21
Implementation issues
Consider the ETD
� y T z
functions
� y � z � 6� � 3·4 +6 � � y T [ � z � 6� � y T z � 6�4 �T ¸6 � for
R s � ° � - - - -
A straightforward implementation suffers from cancellation errors (Kassam andTrefethen).
Numerical techniques
�
Decomposition methods�
Krylov subspace approximations�
Cauchy integral approach
Exponential Integrators and Lie group methods – p.14/21
Cauchy integral approach
Based on the Cauchy integral formula
� y T z � � � +sY¹ R º �
� y T z � { � � { I4 � �� � ) { �
where
»� is a contour in the complex plane that encloses the eigenvalue of
�
, and it isalso well separated from
. It is practical to choose the contour
» � to be a circlecentered on the real axis.Using the trapezoid rule, we obtain the following approximation
� y T z � � �Z¼ +=
<_ � �
{ _ � y T z � { _ � � { _ I4 � �� � �
where
=
is the number of the equally spaced points
{ _ along the contour
»� .
Toachieve computational savings we can use the formula
where the contour encloses the eigenvalues of and is well separated fromfor all in the integration process.As before
where now are the equally spaced points along the contour .
When arises from a finite difference approximation, we can benefit from its sparseblock structure and find the action of the inverse martices to a given vector by:
iterative methods - preconditioned conjugate gradient and multigrid methods.
direct methods - , , , factorization.
Exponential Integrators and Lie group methods – p.15/21
Cauchy integral approach
To achieve computational savings we can use the formula
� y T z � � � � y T z � �U � � +sY¹ R º � y T z � � { � � { I4 U � �� � ) { �
where the contour
»
encloses the eigenvalues of �U �and � »
is well separated from
for all � in the integration process.As before
� y T z � � �¼ +=
<_ � �
{ _ � y T z � � { _ � � { _ I4 U � �� � �
where now
{ _ are the equally spaced points along the contour
»
.
Note: The inverse matrices no longer depend of �.
To achieve computational savingswe can use the formula
where the contour encloses the eigenvalues of and is well separated fromfor all in the integration process.As before
where now are the equally spaced points along the contour .
When arises from a finite difference approximation, we can benefit from its sparseblock structure and find the action of the inverse martices to a given vector by:
iterative methods - preconditioned conjugate gradient and multigrid methods.
direct methods - , , , factorization.
Exponential Integrators and Lie group methods – p.15/21
Cauchy integral approach
To achieve computational savings we can use the formula
� y T z � � � � y T z � �U � � +sY¹ R º � y T z � � { � � { I4 U � �� � ) { �
where the contour
»
encloses the eigenvalues of �U �and � »
is well separated from
for all � in the integration process.As before
� y T z � � �¼ +=
<_ � �
{ _ � y T z � � { _ � � { _ I4 U � �� � �
where now
{ _ are the equally spaced points along the contour
»
.
When
�
arises from a finite difference approximation, we can benefit from its sparseblock structure and find the action of the inverse martices to a given vector by:t iterative methods - preconditioned conjugate gradient and multigrid methods.
t direct methods -½¾
,
Q Q
,
Q � ½¾
,
� S
factorization.
Exponential Integrators and Lie group methods – p.15/21
Numerical experiments
The methodst ETD RK4(Kr) The fourth order method of Krogstad;t IF RK4 The fourth order Integrating Factor Runge–Kutta methd (classical RK);t CF4 The fourth order Commutator Free Lie group method with affine action;t CF4A1 A fourth order CF method with action corresponding to nonautonomous FVF.
Exponential Integrators and Lie group methods – p.16/21
Kuramoto-Sivashinsky equation
P� 4 P P]¿ 4 P]¿ ¿ 4 P]¿ ¿ ¿ ¿ � À, � ° sY¹ �with periodic boundary conditions and with the initial condition
P � À � � ÁÂÃ � À+ ¯ � � + 0Ã ÄYÅ � À+ ¯ � � -
Exponential Integrators and Lie group methods – p.17/21
Kuramoto-Sivashinsky equation
P� 4 P P]¿ 4 P]¿ ¿ 4 P]¿ ¿ ¿ ¿ � À, � ° sY¹ �with periodic boundary conditions and with the initial condition
P � À � � ÁÂÃ � À+ ¯ � � + 0Ã ÄYÅ � À+ ¯ � � -
We discretise the spatial part using Fourier spectral method.The transformed equation in the Fourier space is
Æ P� 4 R=s ÆP � 0 �= � 4 = g � Æ P �
�Ç Æ P � �= � �= � 4 = g � Æ P � = �
and
È � Æ P � � � 4 T <� � Q � � Q� � � Æ P � � � � �
Exponential Integrators and Lie group methods – p.17/21
Kuramoto-Sivashinsky equation
10−2 10−1 10010−7
10−6
10−5
10−4
10−3
10−2
10−1
100
101
Stepsize h
Rel
ativ
e er
ror a
t tim
e =
65 Kuramoto−Sivashinsky equation
IF RK4ETD RK4(Kr)CF4CF4A1
Exponential Integrators and Lie group methods – p.17/21
Allen-Cahn equation
P� É P]¿ ¿ 0 P4 P ¢ � À, 4 + � + �with É - + and with boundary and initial conditions
P �4 + � � � 4 + � P �+ � � � + � P � À � � - Ê° À 0 - ËÌ Ã ÄYÅ �4 + - ʹ À �
Exponential Integrators and Lie group methods – p.18/21
Allen-Cahn equation
P� É P]¿ ¿ 0 P4 P ¢ � À, 4 + � + �with É - + and with boundary and initial conditions
P �4 + � � � 4 + � P �+ � � � + � P � À � � - Ê° À 0 - ËÌ Ã ÄYÅ �4 + - ʹ À �
After discretisation in space.
P� Ç P 0 È � P � � � �
where
Ç É Í � � È � P � � � � P4 P ¢ and
Íis the Chebyshev differentiation matrix.
Exponential Integrators and Lie group methods – p.18/21
Allen-Cahn equation
10−2 10−1 10010−10
10−9
10−8
10−7
10−6
10−5
10−4
10−3
10−2
Stepsize h
Rel
ativ
e er
ror a
t tim
e =
31 Allen−Cahn equation
IF RK4
ETD RK4(Kr)
CF4
CF4A4
Exponential Integrators and Lie group methods – p.18/21
Conclusions
� Studied the connection between exponential integrators andLie group methods.
� Proposed how to construct exponential integrators withalgebra action arising from the solutions of differentialequations with nonautonomous frozen vector fields.
� The proposed approach is promising for non-diagonalexamples
Exponential Integrators and Lie group methods – p.19/21
Future work
� Study the connections between the order of the action andthe order of the methods.
� How to find a good algebra action?
� Stability analysis.
� Extensive numerical experiments.
Exponential Integrators and Lie group methods – p.20/21
References
�
P. Crouch and R. Grossman, Numerical Integration of ordinary differentialequations on manifolds, J. Nonlinear. Sci. 3, 1-33, (1993)�
H. Munthe-Kaas, High order Runge–Kutta methods on manifolds, Appl. Num.Math., 29, 115-127 (1999)�
A. Iserles, H. Munthe-Kaas, S.P. Nørsett and A. Zanna, Lie-group methods, ActaNumerica 9, 215-365, (2000)�
E. Celledoni, A. Marthinsen, B. Owren, Commutator-free lie group methods, FGCS
19(3), 341-352 (2003)�
S. Krogstad, Generalized integrating factor methods for stiff PDEs, available at:http://www.ii.uib.no/ Îstein�
B. Minchev, Exponential integrators for semilinear problems, PhD thesis, availableat: http://www.ii.uib.no/ Îborko
Exponential Integrators and Lie group methods – p.21/21