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Exponential Integrators and Lie group methods Workshop on Exponential Integrators October 20-23, 2004, Innsbruck. Borislav V. Minchev [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

Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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Page 1: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 2: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 3: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 4: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 5: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 6: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 7: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 8: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 9: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 10: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 11: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 12: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 13: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 14: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 15: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 16: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 17: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 18: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 19: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 20: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 21: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 22: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 23: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 24: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 25: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 26: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 27: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 28: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 29: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 30: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 31: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 32: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 33: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 34: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 35: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 36: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 37: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 38: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 39: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 40: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 41: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 42: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 43: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 44: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 45: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 46: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 47: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 48: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 49: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 50: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

Page 51: Exponential Integrators and Lie group methodsborko/pub/insb.pdf · Exponential Integrators and Lie group methods Œ p.10/21. Basic motions on Consider the following nonautonomous

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

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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

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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

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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

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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

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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